mathlib documentation

analysis.calculus.times_cont_diff

Higher differentiability

A function is C^1 on a domain if it is differentiable there, and its derivative is continuous. By induction, it is C^n if it is C^{n-1} and its (n-1)-th derivative is C^1 there or, equivalently, if it is C^1 and its derivative is C^{n-1}. Finally, it is C^∞ if it is C^n for all n.

We formalize these notions by defining iteratively the n+1-th derivative of a function as the derivative of the n-th derivative. It is called iterated_fderiv π•œ n f x where π•œ is the field, n is the number of iterations, f is the function and x is the point, and it is given as an n-multilinear map. We also define a version iterated_fderiv_within relative to a domain, as well as predicates times_cont_diff_within_at, times_cont_diff_at, times_cont_diff_on and times_cont_diff saying that the function is C^n within a set at a point, at a point, on a set and on the whole space respectively.

To avoid the issue of choice when choosing a derivative in sets where the derivative is not necessarily unique, times_cont_diff_on is not defined directly in terms of the regularity of the specific choice iterated_fderiv_within π•œ n f s inside s, but in terms of the existence of a nice sequence of derivatives, expressed with a predicate has_ftaylor_series_up_to_on.

We prove basic properties of these notions.

Main definitions and results

Let f : E β†’ F be a map between normed vector spaces over a nondiscrete normed field π•œ.

In sets of unique differentiability, times_cont_diff_on π•œ n f s can be expressed in terms of the properties of iterated_fderiv_within π•œ m f s for m ≀ n. In the whole space, times_cont_diff π•œ n f can be expressed in terms of the properties of iterated_fderiv π•œ m f for m ≀ n.

We also prove that the usual operations (addition, multiplication, difference, composition, and so on) preserve C^n functions.

Implementation notes

The definitions in this file are designed to work on any field π•œ. They are sometimes slightly more complicated than the naive definitions one would guess from the intuition over the real or complex numbers, but they are designed to circumvent the lack of gluing properties and partitions of unity in general. In the usual situations, they coincide with the usual definitions.

Definition of C^n functions in domains

One could define C^n functions in a domain s by fixing an arbitrary choice of derivatives (this is what we do with iterated_fderiv_within) and requiring that all these derivatives up to n are continuous. If the derivative is not unique, this could lead to strange behavior like two C^n functions f and g on s whose sum is not C^n. A better definition is thus to say that a function is C^n inside s if it admits a sequence of derivatives up to n inside s.

This definition still has the problem that a function which is locally C^n would not need to be C^n, as different choices of sequences of derivatives around different points might possibly not be glued together to give a globally defined sequence of derivatives. (Note that this issue can not happen over reals, thanks to partition of unity, but the behavior over a general field is not so clear, and we want a definition for general fields). Also, there are locality problems for the order parameter: one could image a function which, for each n, has a nice sequence of derivatives up to order n, but they do not coincide for varying n and can therefore not be glued to give rise to an infinite sequence of derivatives. This would give a function which is C^n for all n, but not C^∞. We solve this issue by putting locality conditions in space and order in our definition of times_cont_diff_within_at and times_cont_diff_on. The resulting definition is slightly more complicated to work with (in fact not so much), but it gives rise to completely satisfactory theorems.

For instance, with this definition, a real function which is C^m (but not better) on (-1/m, 1/m) for each natural m is by definition C^∞ at 0.

There is another issue with the definition of times_cont_diff_within_at π•œ n f s x. We can require the existence and good behavior of derivatives up to order n on a neighborhood of x within s. However, this does not imply continuity or differentiability within s of the function at x when x does not belong to s. Therefore, we require such existence and good behavior on a neighborhood of x within s βˆͺ {x} (which appears as insert x s in this file).

Side of the composition, and universe issues

With a naΓ―ve direct definition, the n-th derivative of a function belongs to the space E β†’L[π•œ] (E β†’L[π•œ] (E ... F)...))) where there are n iterations of E β†’L[π•œ]. This space may also be seen as the space of continuous multilinear functions on n copies of E with values in F, by uncurrying. This is the point of view that is usually adopted in textbooks, and that we also use. This means that the definition and the first proofs are slightly involved, as one has to keep track of the uncurrying operation. The uncurrying can be done from the left or from the right, amounting to defining the n+1-th derivative either as the derivative of the n-th derivative, or as the n-th derivative of the derivative. For proofs, it would be more convenient to use the latter approach (from the right), as it means to prove things at the n+1-th step we only need to understand well enough the derivative in E β†’L[π•œ] F (contrary to the approach from the left, where one would need to know enough on the n-th derivative to deduce things on the n+1-th derivative).

However, the definition from the right leads to a universe polymorphism problem: if we define iterated_fderiv π•œ (n + 1) f x = iterated_fderiv π•œ n (fderiv π•œ f) x by induction, we need to generalize over all spaces (as f and fderiv π•œ f don't take values in the same space). It is only possible to generalize over all spaces in some fixed universe in an inductive definition. For f : E β†’ F, then fderiv π•œ f is a map E β†’ (E β†’L[π•œ] F). Therefore, the definition will only work if F and E β†’L[π•œ] F are in the same universe.

This issue does not appear with the definition from the left, where one does not need to generalize over all spaces. Therefore, we use the definition from the left. This means some proofs later on become a little bit more complicated: to prove that a function is C^n, the most efficient approach is to exhibit a formula for its n-th derivative and prove it is continuous (contrary to the inductive approach where one would prove smoothness statements without giving a formula for the derivative). In the end, this approach is still satisfactory as it is good to have formulas for the iterated derivatives in various constructions.

One point where we depart from this explicit approach is in the proof of smoothness of a composition: there is a formula for the n-th derivative of a composition (FaΓ  di Bruno's formula), but it is very complicated and barely usable, while the inductive proof is very simple. Thus, we give the inductive proof. As explained above, it works by generalizing over the target space, hence it only works well if all spaces belong to the same universe. To get the general version, we lift things to a common universe using a trick.

Variables management

The textbook definitions and proofs use various identifications and abuse of notations, for instance when saying that the natural space in which the derivative lives, i.e., E β†’L[π•œ] (E β†’L[π•œ] ( ... β†’L[π•œ] F)), is the same as a space of multilinear maps. When doing things formally, we need to provide explicit maps for these identifications, and chase some diagrams to see everything is compatible with the identifications. In particular, one needs to check that taking the derivative and then doing the identification, or first doing the identification and then taking the derivative, gives the same result. The key point for this is that taking the derivative commutes with continuous linear equivalences. Therefore, we need to implement all our identifications with continuous linear equivs.

Notations

We use the notation E [Γ—n]β†’L[π•œ] F for the space of continuous multilinear maps on E^n with values in F. This is the space in which the n-th derivative of a function from E to F lives.

In this file, we denote ⊀ : with_top β„• with ∞.

Tags

derivative, differentiability, higher derivative, C^n, multilinear, Taylor series, formal series

def formal_multilinear_series (π•œ : Type u_1) [nondiscrete_normed_field π•œ] (E : Type u_2) [normed_group E] [normed_space π•œ E] (F : Type u_3) [normed_group F] [normed_space π•œ F] :
Type (max u_2 u_3)

A formal multilinear series over a field π•œ, from E to F, is given by a family of multilinear maps from E^n to F for all n.

Equations
@[instance]
def formal_multilinear_series.add_comm_group (π•œ : Type u_1) [nondiscrete_normed_field π•œ] (E : Type u_2) [normed_group E] [normed_space π•œ E] (F : Type u_3) [normed_group F] [normed_space π•œ F] :

@[instance]
def formal_multilinear_series.inhabited {π•œ : Type u_1} [nondiscrete_normed_field π•œ] {E : Type u_2} [normed_group E] [normed_space π•œ E] {F : Type u_3} [normed_group F] [normed_space π•œ F] :

Equations
@[instance]
def formal_multilinear_series.module {π•œ : Type u_1} [nondiscrete_normed_field π•œ] {E : Type u_2} [normed_group E] [normed_space π•œ E] {F : Type u_3} [normed_group F] [normed_space π•œ F] :
module π•œ (formal_multilinear_series π•œ E F)

Equations
def formal_multilinear_series.shift {π•œ : Type u_1} [nondiscrete_normed_field π•œ] {E : Type u_2} [normed_group E] [normed_space π•œ E] {F : Type u_3} [normed_group F] [normed_space π•œ F] :
formal_multilinear_series π•œ E F β†’ formal_multilinear_series π•œ E (E β†’L[π•œ] F)

Forgetting the zeroth term in a formal multilinear series, and interpreting the following terms as multilinear maps into E β†’L[π•œ] F. If p corresponds to the Taylor series of a function, then p.shift is the Taylor series of the derivative of the function.

Equations
def formal_multilinear_series.unshift {π•œ : Type u_1} [nondiscrete_normed_field π•œ] {E : Type u_2} [normed_group E] [normed_space π•œ E] {F : Type u_3} [normed_group F] [normed_space π•œ F] :
formal_multilinear_series π•œ E (E β†’L[π•œ] F) β†’ F β†’ formal_multilinear_series π•œ E F

Adding a zeroth term to a formal multilinear series taking values in E β†’L[π•œ] F. This corresponds to starting from a Taylor series for the derivative of a function, and building a Taylor series for the function itself.

Equations
theorem formal_multilinear_series.congr {π•œ : Type u_1} [nondiscrete_normed_field π•œ] {E : Type u_2} [normed_group E] [normed_space π•œ E] {F : Type u_3} [normed_group F] [normed_space π•œ F] (p : formal_multilinear_series π•œ E F) {m n : β„•} {v : fin m β†’ E} {w : fin n β†’ E} :
m = n β†’ (βˆ€ (i : β„•) (him : i < m) (hin : i < n), v ⟨i, him⟩ = w ⟨i, hin⟩) β†’ ⇑(p m) v = ⇑(p n) w

Convenience congruence lemma stating in a dependent setting that, if the arguments to a formal multilinear series are equal, then the values are also equal.

Functions with a Taylor series on a domain

structure has_ftaylor_series_up_to_on {π•œ : Type u_1} [nondiscrete_normed_field π•œ] {E : Type u_2} [normed_group E] [normed_space π•œ E] {F : Type u_3} [normed_group F] [normed_space π•œ F] :
with_top β„• β†’ (E β†’ F) β†’ (E β†’ formal_multilinear_series π•œ E F) β†’ set E β†’ Prop

has_ftaylor_series_up_to_on n f p s registers the fact that p 0 = f and p (m+1) is a derivative of p m for m < n, and is continuous for m ≀ n. This is a predicate analogous to has_fderiv_within_at but for higher order derivatives.

theorem has_ftaylor_series_up_to_on.zero_eq' {π•œ : Type u_1} [nondiscrete_normed_field π•œ] {E : Type u_2} [normed_group E] [normed_space π•œ E] {F : Type u_3} [normed_group F] [normed_space π•œ F] {s : set E} {f : E β†’ F} {p : E β†’ formal_multilinear_series π•œ E F} {n : with_top β„•} (h : has_ftaylor_series_up_to_on n f p s) {x : E} :
x ∈ s β†’ p x 0 = ⇑((continuous_multilinear_curry_fin0 π•œ E F).symm) (f x)

theorem has_ftaylor_series_up_to_on.congr {π•œ : Type u_1} [nondiscrete_normed_field π•œ] {E : Type u_2} [normed_group E] [normed_space π•œ E] {F : Type u_3} [normed_group F] [normed_space π•œ F] {s : set E} {f f₁ : E β†’ F} {p : E β†’ formal_multilinear_series π•œ E F} {n : with_top β„•} :
has_ftaylor_series_up_to_on n f p s β†’ (βˆ€ (x : E), x ∈ s β†’ f₁ x = f x) β†’ has_ftaylor_series_up_to_on n f₁ p s

If two functions coincide on a set s, then a Taylor series for the first one is as well a Taylor series for the second one.

theorem has_ftaylor_series_up_to_on.mono {π•œ : Type u_1} [nondiscrete_normed_field π•œ] {E : Type u_2} [normed_group E] [normed_space π•œ E] {F : Type u_3} [normed_group F] [normed_space π•œ F] {s : set E} {f : E β†’ F} {p : E β†’ formal_multilinear_series π•œ E F} {n : with_top β„•} (h : has_ftaylor_series_up_to_on n f p s) {t : set E} :

theorem has_ftaylor_series_up_to_on.of_le {π•œ : Type u_1} [nondiscrete_normed_field π•œ] {E : Type u_2} [normed_group E] [normed_space π•œ E] {F : Type u_3} [normed_group F] [normed_space π•œ F] {s : set E} {f : E β†’ F} {p : E β†’ formal_multilinear_series π•œ E F} {m n : with_top β„•} :

theorem has_ftaylor_series_up_to_on.continuous_on {π•œ : Type u_1} [nondiscrete_normed_field π•œ] {E : Type u_2} [normed_group E] [normed_space π•œ E] {F : Type u_3} [normed_group F] [normed_space π•œ F] {s : set E} {f : E β†’ F} {p : E β†’ formal_multilinear_series π•œ E F} {n : with_top β„•} :

theorem has_ftaylor_series_up_to_on_zero_iff {π•œ : Type u_1} [nondiscrete_normed_field π•œ] {E : Type u_2} [normed_group E] [normed_space π•œ E] {F : Type u_3} [normed_group F] [normed_space π•œ F] {s : set E} {f : E β†’ F} {p : E β†’ formal_multilinear_series π•œ E F} :
has_ftaylor_series_up_to_on 0 f p s ↔ continuous_on f s ∧ βˆ€ (x : E), x ∈ s β†’ (p x 0).uncurry0 = f x

theorem has_ftaylor_series_up_to_on_top_iff {π•œ : Type u_1} [nondiscrete_normed_field π•œ] {E : Type u_2} [normed_group E] [normed_space π•œ E] {F : Type u_3} [normed_group F] [normed_space π•œ F] {s : set E} {f : E β†’ F} {p : E β†’ formal_multilinear_series π•œ E F} :

theorem has_ftaylor_series_up_to_on.has_fderiv_within_at {π•œ : Type u_1} [nondiscrete_normed_field π•œ] {E : Type u_2} [normed_group E] [normed_space π•œ E] {F : Type u_3} [normed_group F] [normed_space π•œ F] {s : set E} {f : E β†’ F} {x : E} {p : E β†’ formal_multilinear_series π•œ E F} {n : with_top β„•} :
has_ftaylor_series_up_to_on n f p s β†’ 1 ≀ n β†’ x ∈ s β†’ has_fderiv_within_at f (⇑(continuous_multilinear_curry_fin1 π•œ E F) (p x 1)) s x

If a function has a Taylor series at order at least 1, then the term of order 1 of this series is a derivative of f.

theorem has_ftaylor_series_up_to_on.differentiable_on {π•œ : Type u_1} [nondiscrete_normed_field π•œ] {E : Type u_2} [normed_group E] [normed_space π•œ E] {F : Type u_3} [normed_group F] [normed_space π•œ F] {s : set E} {f : E β†’ F} {p : E β†’ formal_multilinear_series π•œ E F} {n : with_top β„•} :
has_ftaylor_series_up_to_on n f p s β†’ 1 ≀ n β†’ differentiable_on π•œ f s

theorem has_ftaylor_series_up_to_on_succ_iff_left {π•œ : Type u_1} [nondiscrete_normed_field π•œ] {E : Type u_2} [normed_group E] [normed_space π•œ E] {F : Type u_3} [normed_group F] [normed_space π•œ F] {s : set E} {f : E β†’ F} {p : E β†’ formal_multilinear_series π•œ E F} {n : β„•} :
has_ftaylor_series_up_to_on (↑n + 1) f p s ↔ has_ftaylor_series_up_to_on ↑n f p s ∧ (βˆ€ (x : E), x ∈ s β†’ has_fderiv_within_at (Ξ» (y : E), p y n) (p x n.succ).curry_left s x) ∧ continuous_on (Ξ» (x : E), p x (n + 1)) s

p is a Taylor series of f up to n+1 if and only if p is a Taylor series up to n, and p (n + 1) is a derivative of p n.

theorem has_ftaylor_series_up_to_on_succ_iff_right {π•œ : Type u_1} [nondiscrete_normed_field π•œ] {E : Type u_2} [normed_group E] [normed_space π•œ E] {F : Type u_3} [normed_group F] [normed_space π•œ F] {s : set E} {f : E β†’ F} {p : E β†’ formal_multilinear_series π•œ E F} {n : β„•} :
has_ftaylor_series_up_to_on ↑(n + 1) f p s ↔ (βˆ€ (x : E), x ∈ s β†’ (p x 0).uncurry0 = f x) ∧ (βˆ€ (x : E), x ∈ s β†’ has_fderiv_within_at (Ξ» (y : E), p y 0) (p x 1).curry_left s x) ∧ has_ftaylor_series_up_to_on ↑n (Ξ» (x : E), ⇑(continuous_multilinear_curry_fin1 π•œ E F) (p x 1)) (Ξ» (x : E), (p x).shift) s

p is a Taylor series of f up to n+1 if and only if p.shift is a Taylor series up to n for p 1, which is a derivative of f.

Smooth functions within a set around a point

def times_cont_diff_within_at (π•œ : Type u_1) [nondiscrete_normed_field π•œ] {E : Type u_2} [normed_group E] [normed_space π•œ E] {F : Type u_3} [normed_group F] [normed_space π•œ F] :
with_top β„• β†’ (E β†’ F) β†’ set E β†’ E β†’ Prop

A function is continuously differentiable up to order n within a set s at a point x if it admits continuous derivatives up to order n in a neighborhood of x in s βˆͺ {x}. For n = ∞, we only require that this holds up to any finite order (where the neighborhood may depend on the finite order we consider).

For instance, a real function which is C^m on (-1/m, 1/m) for each natural m, but not better, is C^∞ at 0 within univ.

Equations
theorem times_cont_diff_within_at_nat {π•œ : Type u_1} [nondiscrete_normed_field π•œ] {E : Type u_2} [normed_group E] [normed_space π•œ E] {F : Type u_3} [normed_group F] [normed_space π•œ F] {s : set E} {f : E β†’ F} {x : E} {n : β„•} :
times_cont_diff_within_at π•œ ↑n f s x ↔ βˆƒ (u : set E) (H : u ∈ 𝓝[insert x s] x) (p : E β†’ formal_multilinear_series π•œ E F), has_ftaylor_series_up_to_on ↑n f p u

theorem times_cont_diff_within_at_top {π•œ : Type u_1} [nondiscrete_normed_field π•œ] {E : Type u_2} [normed_group E] [normed_space π•œ E] {F : Type u_3} [normed_group F] [normed_space π•œ F] {s : set E} {f : E β†’ F} {x : E} :
times_cont_diff_within_at π•œ ⊀ f s x ↔ βˆ€ (n : β„•), times_cont_diff_within_at π•œ ↑n f s x

theorem times_cont_diff_within_at.continuous_within_at' {π•œ : Type u_1} [nondiscrete_normed_field π•œ] {E : Type u_2} [normed_group E] [normed_space π•œ E] {F : Type u_3} [normed_group F] [normed_space π•œ F] {s : set E} {f : E β†’ F} {x : E} {n : with_top β„•} :
times_cont_diff_within_at π•œ n f s x β†’ continuous_within_at f (insert x s) x

theorem times_cont_diff_within_at.continuous_within_at {π•œ : Type u_1} [nondiscrete_normed_field π•œ] {E : Type u_2} [normed_group E] [normed_space π•œ E] {F : Type u_3} [normed_group F] [normed_space π•œ F] {s : set E} {f : E β†’ F} {x : E} {n : with_top β„•} :
times_cont_diff_within_at π•œ n f s x β†’ continuous_within_at f s x

theorem times_cont_diff_within_at.congr_of_eventually_eq {π•œ : Type u_1} [nondiscrete_normed_field π•œ] {E : Type u_2} [normed_group E] [normed_space π•œ E] {F : Type u_3} [normed_group F] [normed_space π•œ F] {s : set E} {f f₁ : E β†’ F} {x : E} {n : with_top β„•} :
times_cont_diff_within_at π•œ n f s x β†’ f₁ =αΆ [𝓝[s] x] f β†’ f₁ x = f x β†’ times_cont_diff_within_at π•œ n f₁ s x

theorem times_cont_diff_within_at.congr_of_eventually_eq' {π•œ : Type u_1} [nondiscrete_normed_field π•œ] {E : Type u_2} [normed_group E] [normed_space π•œ E] {F : Type u_3} [normed_group F] [normed_space π•œ F] {s : set E} {f f₁ : E β†’ F} {x : E} {n : with_top β„•} :
times_cont_diff_within_at π•œ n f s x β†’ f₁ =αΆ [𝓝[s] x] f β†’ x ∈ s β†’ times_cont_diff_within_at π•œ n f₁ s x

theorem filter.eventually_eq.times_cont_diff_within_at_iff {π•œ : Type u_1} [nondiscrete_normed_field π•œ] {E : Type u_2} [normed_group E] [normed_space π•œ E] {F : Type u_3} [normed_group F] [normed_space π•œ F] {s : set E} {f f₁ : E β†’ F} {x : E} {n : with_top β„•} :
f₁ =αΆ [𝓝[s] x] f β†’ f₁ x = f x β†’ (times_cont_diff_within_at π•œ n f₁ s x ↔ times_cont_diff_within_at π•œ n f s x)

theorem times_cont_diff_within_at.congr {π•œ : Type u_1} [nondiscrete_normed_field π•œ] {E : Type u_2} [normed_group E] [normed_space π•œ E] {F : Type u_3} [normed_group F] [normed_space π•œ F] {s : set E} {f f₁ : E β†’ F} {x : E} {n : with_top β„•} :
times_cont_diff_within_at π•œ n f s x β†’ (βˆ€ (y : E), y ∈ s β†’ f₁ y = f y) β†’ f₁ x = f x β†’ times_cont_diff_within_at π•œ n f₁ s x

theorem times_cont_diff_within_at.mono {π•œ : Type u_1} [nondiscrete_normed_field π•œ] {E : Type u_2} [normed_group E] [normed_space π•œ E] {F : Type u_3} [normed_group F] [normed_space π•œ F] {s : set E} {f : E β†’ F} {x : E} {n : with_top β„•} (h : times_cont_diff_within_at π•œ n f s x) {t : set E} :
t βŠ† s β†’ times_cont_diff_within_at π•œ n f t x

theorem times_cont_diff_within_at.of_le {π•œ : Type u_1} [nondiscrete_normed_field π•œ] {E : Type u_2} [normed_group E] [normed_space π•œ E] {F : Type u_3} [normed_group F] [normed_space π•œ F] {s : set E} {f : E β†’ F} {x : E} {m n : with_top β„•} :
times_cont_diff_within_at π•œ n f s x β†’ m ≀ n β†’ times_cont_diff_within_at π•œ m f s x

theorem times_cont_diff_within_at_inter' {π•œ : Type u_1} [nondiscrete_normed_field π•œ] {E : Type u_2} [normed_group E] [normed_space π•œ E] {F : Type u_3} [normed_group F] [normed_space π•œ F] {s t : set E} {f : E β†’ F} {x : E} {n : with_top β„•} :
t ∈ 𝓝[s] x β†’ (times_cont_diff_within_at π•œ n f (s ∩ t) x ↔ times_cont_diff_within_at π•œ n f s x)

theorem times_cont_diff_within_at_inter {π•œ : Type u_1} [nondiscrete_normed_field π•œ] {E : Type u_2} [normed_group E] [normed_space π•œ E] {F : Type u_3} [normed_group F] [normed_space π•œ F] {s t : set E} {f : E β†’ F} {x : E} {n : with_top β„•} :
t ∈ 𝓝 x β†’ (times_cont_diff_within_at π•œ n f (s ∩ t) x ↔ times_cont_diff_within_at π•œ n f s x)

theorem times_cont_diff_within_at.differentiable_within_at' {π•œ : Type u_1} [nondiscrete_normed_field π•œ] {E : Type u_2} [normed_group E] [normed_space π•œ E] {F : Type u_3} [normed_group F] [normed_space π•œ F] {s : set E} {f : E β†’ F} {x : E} {n : with_top β„•} :
times_cont_diff_within_at π•œ n f s x β†’ 1 ≀ n β†’ differentiable_within_at π•œ f (insert x s) x

If a function is C^n within a set at a point, with n β‰₯ 1, then it is differentiable within this set at this point.

theorem times_cont_diff_within_at.differentiable_within_at {π•œ : Type u_1} [nondiscrete_normed_field π•œ] {E : Type u_2} [normed_group E] [normed_space π•œ E] {F : Type u_3} [normed_group F] [normed_space π•œ F] {s : set E} {f : E β†’ F} {x : E} {n : with_top β„•} :
times_cont_diff_within_at π•œ n f s x β†’ 1 ≀ n β†’ differentiable_within_at π•œ f s x

theorem times_cont_diff_within_at_succ_iff_has_fderiv_within_at {π•œ : Type u_1} [nondiscrete_normed_field π•œ] {E : Type u_2} [normed_group E] [normed_space π•œ E] {F : Type u_3} [normed_group F] [normed_space π•œ F] {s : set E} {f : E β†’ F} {x : E} {n : β„•} :
times_cont_diff_within_at π•œ ↑(n + 1) f s x ↔ βˆƒ (u : set E) (H : u ∈ 𝓝[insert x s] x) (f' : E β†’ (E β†’L[π•œ] F)), (βˆ€ (x : E), x ∈ u β†’ has_fderiv_within_at f (f' x) u x) ∧ times_cont_diff_within_at π•œ ↑n f' u x

A function is C^(n + 1) on a domain iff locally, it has a derivative which is C^n.

Smooth functions within a set

def times_cont_diff_on (π•œ : Type u_1) [nondiscrete_normed_field π•œ] {E : Type u_2} [normed_group E] [normed_space π•œ E] {F : Type u_3} [normed_group F] [normed_space π•œ F] :
with_top β„• β†’ (E β†’ F) β†’ set E β†’ Prop

A function is continuously differentiable up to n on s if, for any point x in s, it admits continuous derivatives up to order n on a neighborhood of x in s.

For n = ∞, we only require that this holds up to any finite order (where the neighborhood may depend on the finite order we consider).

Equations
theorem times_cont_diff_on.times_cont_diff_within_at {π•œ : Type u_1} [nondiscrete_normed_field π•œ] {E : Type u_2} [normed_group E] [normed_space π•œ E] {F : Type u_3} [normed_group F] [normed_space π•œ F] {s : set E} {f : E β†’ F} {x : E} {n : with_top β„•} :
times_cont_diff_on π•œ n f s β†’ x ∈ s β†’ times_cont_diff_within_at π•œ n f s x

theorem times_cont_diff_within_at.times_cont_diff_on {π•œ : Type u_1} [nondiscrete_normed_field π•œ] {E : Type u_2} [normed_group E] [normed_space π•œ E] {F : Type u_3} [normed_group F] [normed_space π•œ F] {s : set E} {f : E β†’ F} {x : E} {n : with_top β„•} {m : β„•} :
↑m ≀ n β†’ times_cont_diff_within_at π•œ n f s x β†’ (βˆƒ (u : set E) (H : u ∈ 𝓝[insert x s] x), u βŠ† insert x s ∧ times_cont_diff_on π•œ ↑m f u)

theorem times_cont_diff_on_top {π•œ : Type u_1} [nondiscrete_normed_field π•œ] {E : Type u_2} [normed_group E] [normed_space π•œ E] {F : Type u_3} [normed_group F] [normed_space π•œ F] {s : set E} {f : E β†’ F} :
times_cont_diff_on π•œ ⊀ f s ↔ βˆ€ (n : β„•), times_cont_diff_on π•œ ↑n f s

theorem times_cont_diff_on.continuous_on {π•œ : Type u_1} [nondiscrete_normed_field π•œ] {E : Type u_2} [normed_group E] [normed_space π•œ E] {F : Type u_3} [normed_group F] [normed_space π•œ F] {s : set E} {f : E β†’ F} {n : with_top β„•} :
times_cont_diff_on π•œ n f s β†’ continuous_on f s

theorem times_cont_diff_on.congr {π•œ : Type u_1} [nondiscrete_normed_field π•œ] {E : Type u_2} [normed_group E] [normed_space π•œ E] {F : Type u_3} [normed_group F] [normed_space π•œ F] {s : set E} {f f₁ : E β†’ F} {n : with_top β„•} :
times_cont_diff_on π•œ n f s β†’ (βˆ€ (x : E), x ∈ s β†’ f₁ x = f x) β†’ times_cont_diff_on π•œ n f₁ s

theorem times_cont_diff_on_congr {π•œ : Type u_1} [nondiscrete_normed_field π•œ] {E : Type u_2} [normed_group E] [normed_space π•œ E] {F : Type u_3} [normed_group F] [normed_space π•œ F] {s : set E} {f f₁ : E β†’ F} {n : with_top β„•} :
(βˆ€ (x : E), x ∈ s β†’ f₁ x = f x) β†’ (times_cont_diff_on π•œ n f₁ s ↔ times_cont_diff_on π•œ n f s)

theorem times_cont_diff_on.mono {π•œ : Type u_1} [nondiscrete_normed_field π•œ] {E : Type u_2} [normed_group E] [normed_space π•œ E] {F : Type u_3} [normed_group F] [normed_space π•œ F] {s : set E} {f : E β†’ F} {n : with_top β„•} (h : times_cont_diff_on π•œ n f s) {t : set E} :
t βŠ† s β†’ times_cont_diff_on π•œ n f t

theorem times_cont_diff_on.congr_mono {π•œ : Type u_1} [nondiscrete_normed_field π•œ] {E : Type u_2} [normed_group E] [normed_space π•œ E] {F : Type u_3} [normed_group F] [normed_space π•œ F] {s s₁ : set E} {f f₁ : E β†’ F} {n : with_top β„•} :
times_cont_diff_on π•œ n f s β†’ (βˆ€ (x : E), x ∈ s₁ β†’ f₁ x = f x) β†’ s₁ βŠ† s β†’ times_cont_diff_on π•œ n f₁ s₁

theorem times_cont_diff_on.of_le {π•œ : Type u_1} [nondiscrete_normed_field π•œ] {E : Type u_2} [normed_group E] [normed_space π•œ E] {F : Type u_3} [normed_group F] [normed_space π•œ F] {s : set E} {f : E β†’ F} {m n : with_top β„•} :
times_cont_diff_on π•œ n f s β†’ m ≀ n β†’ times_cont_diff_on π•œ m f s

theorem times_cont_diff_on.differentiable_on {π•œ : Type u_1} [nondiscrete_normed_field π•œ] {E : Type u_2} [normed_group E] [normed_space π•œ E] {F : Type u_3} [normed_group F] [normed_space π•œ F] {s : set E} {f : E β†’ F} {n : with_top β„•} :
times_cont_diff_on π•œ n f s β†’ 1 ≀ n β†’ differentiable_on π•œ f s

If a function is C^n on a set with n β‰₯ 1, then it is differentiable there.

theorem times_cont_diff_on_of_locally_times_cont_diff_on {π•œ : Type u_1} [nondiscrete_normed_field π•œ] {E : Type u_2} [normed_group E] [normed_space π•œ E] {F : Type u_3} [normed_group F] [normed_space π•œ F] {s : set E} {f : E β†’ F} {n : with_top β„•} :
(βˆ€ (x : E), x ∈ s β†’ (βˆƒ (u : set E), is_open u ∧ x ∈ u ∧ times_cont_diff_on π•œ n f (s ∩ u))) β†’ times_cont_diff_on π•œ n f s

If a function is C^n around each point in a set, then it is C^n on the set.

theorem times_cont_diff_on_succ_iff_has_fderiv_within_at {π•œ : Type u_1} [nondiscrete_normed_field π•œ] {E : Type u_2} [normed_group E] [normed_space π•œ E] {F : Type u_3} [normed_group F] [normed_space π•œ F] {s : set E} {f : E β†’ F} {n : β„•} :
times_cont_diff_on π•œ ↑(n + 1) f s ↔ βˆ€ (x : E), x ∈ s β†’ (βˆƒ (u : set E) (H : u ∈ 𝓝[insert x s] x) (f' : E β†’ (E β†’L[π•œ] F)), (βˆ€ (x : E), x ∈ u β†’ has_fderiv_within_at f (f' x) u x) ∧ times_cont_diff_on π•œ ↑n f' u)

A function is C^(n + 1) on a domain iff locally, it has a derivative which is C^n.

Iterated derivative within a set

def iterated_fderiv_within (π•œ : Type u_1) [nondiscrete_normed_field π•œ] {E : Type u_2} [normed_group E] [normed_space π•œ E] {F : Type u_3} [normed_group F] [normed_space π•œ F] (n : β„•) :
(E β†’ F) β†’ set E β†’ E β†’ continuous_multilinear_map π•œ (Ξ» (i : fin n), E) F

The n-th derivative of a function along a set, defined inductively by saying that the n+1-th derivative of f is the derivative of the n-th derivative of f along this set, together with an uncurrying step to see it as a multilinear map in n+1 variables..

Equations
def ftaylor_series_within (π•œ : Type u_1) [nondiscrete_normed_field π•œ] {E : Type u_2} [normed_group E] [normed_space π•œ E] {F : Type u_3} [normed_group F] [normed_space π•œ F] :
(E β†’ F) β†’ set E β†’ E β†’ formal_multilinear_series π•œ E F

Formal Taylor series associated to a function within a set.

Equations
@[simp]
theorem iterated_fderiv_within_zero_apply {π•œ : Type u_1} [nondiscrete_normed_field π•œ] {E : Type u_2} [normed_group E] [normed_space π•œ E] {F : Type u_3} [normed_group F] [normed_space π•œ F] {s : set E} {f : E β†’ F} {x : E} (m : fin 0 β†’ E) :
⇑(iterated_fderiv_within π•œ 0 f s x) m = f x

theorem iterated_fderiv_within_zero_eq_comp {π•œ : Type u_1} [nondiscrete_normed_field π•œ] {E : Type u_2} [normed_group E] [normed_space π•œ E] {F : Type u_3} [normed_group F] [normed_space π•œ F] {s : set E} {f : E β†’ F} :

theorem iterated_fderiv_within_succ_apply_left {π•œ : Type u_1} [nondiscrete_normed_field π•œ] {E : Type u_2} [normed_group E] [normed_space π•œ E] {F : Type u_3} [normed_group F] [normed_space π•œ F] {s : set E} {f : E β†’ F} {x : E} {n : β„•} (m : fin (n + 1) β†’ E) :
⇑(iterated_fderiv_within π•œ (n + 1) f s x) m = ⇑(⇑(fderiv_within π•œ (iterated_fderiv_within π•œ n f s) s x) (m 0)) (fin.tail m)

theorem iterated_fderiv_within_succ_eq_comp_left {π•œ : Type u_1} [nondiscrete_normed_field π•œ] {E : Type u_2} [normed_group E] [normed_space π•œ E] {F : Type u_3} [normed_group F] [normed_space π•œ F] {s : set E} {f : E β†’ F} {n : β„•} :
iterated_fderiv_within π•œ (n + 1) f s = ⇑(continuous_multilinear_curry_left_equiv π•œ (Ξ» (i : fin (n + 1)), E) F) ∘ fderiv_within π•œ (iterated_fderiv_within π•œ n f s) s

Writing explicitly the n+1-th derivative as the composition of a currying linear equiv, and the derivative of the n-th derivative.

theorem iterated_fderiv_within_succ_apply_right {π•œ : Type u_1} [nondiscrete_normed_field π•œ] {E : Type u_2} [normed_group E] [normed_space π•œ E] {F : Type u_3} [normed_group F] [normed_space π•œ F] {s : set E} {f : E β†’ F} {x : E} {n : β„•} (hs : unique_diff_on π•œ s) (hx : x ∈ s) (m : fin (n + 1) β†’ E) :
⇑(iterated_fderiv_within π•œ (n + 1) f s x) m = ⇑(⇑(iterated_fderiv_within π•œ n (Ξ» (y : E), fderiv_within π•œ f s y) s x) (fin.init m)) (m (fin.last n))

theorem iterated_fderiv_within_succ_eq_comp_right {π•œ : Type u_1} [nondiscrete_normed_field π•œ] {E : Type u_2} [normed_group E] [normed_space π•œ E] {F : Type u_3} [normed_group F] [normed_space π•œ F] {s : set E} {f : E β†’ F} {x : E} {n : β„•} :
unique_diff_on π•œ s β†’ x ∈ s β†’ iterated_fderiv_within π•œ (n + 1) f s x = (⇑(continuous_multilinear_curry_right_equiv π•œ (Ξ» (i : fin (n + 1)), E) F) ∘ iterated_fderiv_within π•œ n (Ξ» (y : E), fderiv_within π•œ f s y) s) x

Writing explicitly the n+1-th derivative as the composition of a currying linear equiv, and the n-th derivative of the derivative.

@[simp]
theorem iterated_fderiv_within_one_apply {π•œ : Type u_1} [nondiscrete_normed_field π•œ] {E : Type u_2} [normed_group E] [normed_space π•œ E] {F : Type u_3} [normed_group F] [normed_space π•œ F] {s : set E} {f : E β†’ F} {x : E} (hs : unique_diff_on π•œ s) (hx : x ∈ s) (m : fin 1 β†’ E) :
⇑(iterated_fderiv_within π•œ 1 f s x) m = ⇑(fderiv_within π•œ f s x) (m 0)

theorem iterated_fderiv_within_congr {π•œ : Type u_1} [nondiscrete_normed_field π•œ] {E : Type u_2} [normed_group E] [normed_space π•œ E] {F : Type u_3} [normed_group F] [normed_space π•œ F] {s : set E} {f f₁ : E β†’ F} {x : E} {n : β„•} :
unique_diff_on π•œ s β†’ (βˆ€ (y : E), y ∈ s β†’ f₁ y = f y) β†’ x ∈ s β†’ iterated_fderiv_within π•œ n f₁ s x = iterated_fderiv_within π•œ n f s x

If two functions coincide on a set s of unique differentiability, then their iterated differentials within this set coincide.

theorem iterated_fderiv_within_inter_open {π•œ : Type u_1} [nondiscrete_normed_field π•œ] {E : Type u_2} [normed_group E] [normed_space π•œ E] {F : Type u_3} [normed_group F] [normed_space π•œ F] {s u : set E} {f : E β†’ F} {x : E} {n : β„•} :
is_open u β†’ unique_diff_on π•œ (s ∩ u) β†’ x ∈ s ∩ u β†’ iterated_fderiv_within π•œ n f (s ∩ u) x = iterated_fderiv_within π•œ n f s x

The iterated differential within a set s at a point x is not modified if one intersects s with an open set containing x.

theorem iterated_fderiv_within_inter' {π•œ : Type u_1} [nondiscrete_normed_field π•œ] {E : Type u_2} [normed_group E] [normed_space π•œ E] {F : Type u_3} [normed_group F] [normed_space π•œ F] {s u : set E} {f : E β†’ F} {x : E} {n : β„•} :
u ∈ 𝓝[s] x β†’ unique_diff_on π•œ s β†’ x ∈ s β†’ iterated_fderiv_within π•œ n f (s ∩ u) x = iterated_fderiv_within π•œ n f s x

The iterated differential within a set s at a point x is not modified if one intersects s with a neighborhood of x within s.

theorem iterated_fderiv_within_inter {π•œ : Type u_1} [nondiscrete_normed_field π•œ] {E : Type u_2} [normed_group E] [normed_space π•œ E] {F : Type u_3} [normed_group F] [normed_space π•œ F] {s u : set E} {f : E β†’ F} {x : E} {n : β„•} :
u ∈ 𝓝 x β†’ unique_diff_on π•œ s β†’ x ∈ s β†’ iterated_fderiv_within π•œ n f (s ∩ u) x = iterated_fderiv_within π•œ n f s x

The iterated differential within a set s at a point x is not modified if one intersects s with a neighborhood of x.

@[simp]
theorem times_cont_diff_on_zero {π•œ : Type u_1} [nondiscrete_normed_field π•œ] {E : Type u_2} [normed_group E] [normed_space π•œ E] {F : Type u_3} [normed_group F] [normed_space π•œ F] {s : set E} {f : E β†’ F} :

theorem times_cont_diff_within_at_zero {π•œ : Type u_1} [nondiscrete_normed_field π•œ] {E : Type u_2} [normed_group E] [normed_space π•œ E] {F : Type u_3} [normed_group F] [normed_space π•œ F] {s : set E} {f : E β†’ F} {x : E} :
x ∈ s β†’ (times_cont_diff_within_at π•œ 0 f s x ↔ βˆƒ (u : set E) (H : u ∈ 𝓝[s] x), continuous_on f (s ∩ u))

theorem has_ftaylor_series_up_to_on.eq_ftaylor_series_of_unique_diff_on {π•œ : Type u_1} [nondiscrete_normed_field π•œ] {E : Type u_2} [normed_group E] [normed_space π•œ E] {F : Type u_3} [normed_group F] [normed_space π•œ F] {s : set E} {f : E β†’ F} {x : E} {p : E β†’ formal_multilinear_series π•œ E F} {n : with_top β„•} (h : has_ftaylor_series_up_to_on n f p s) {m : β„•} :
↑m ≀ n β†’ unique_diff_on π•œ s β†’ x ∈ s β†’ p x m = iterated_fderiv_within π•œ m f s x

On a set with unique differentiability, any choice of iterated differential has to coincide with the one we have chosen in iterated_fderiv_within π•œ m f s.

theorem times_cont_diff_on.ftaylor_series_within {π•œ : Type u_1} [nondiscrete_normed_field π•œ] {E : Type u_2} [normed_group E] [normed_space π•œ E] {F : Type u_3} [normed_group F] [normed_space π•œ F] {s : set E} {f : E β†’ F} {n : with_top β„•} :
times_cont_diff_on π•œ n f s β†’ unique_diff_on π•œ s β†’ has_ftaylor_series_up_to_on n f (ftaylor_series_within π•œ f s) s

When a function is C^n in a set s of unique differentiability, it admits ftaylor_series_within π•œ f s as a Taylor series up to order n in s.

theorem times_cont_diff_on_of_continuous_on_differentiable_on {π•œ : Type u_1} [nondiscrete_normed_field π•œ] {E : Type u_2} [normed_group E] [normed_space π•œ E] {F : Type u_3} [normed_group F] [normed_space π•œ F] {s : set E} {f : E β†’ F} {n : with_top β„•} :
(βˆ€ (m : β„•), ↑m ≀ n β†’ continuous_on (Ξ» (x : E), iterated_fderiv_within π•œ m f s x) s) β†’ (βˆ€ (m : β„•), ↑m < n β†’ differentiable_on π•œ (Ξ» (x : E), iterated_fderiv_within π•œ m f s x) s) β†’ times_cont_diff_on π•œ n f s

theorem times_cont_diff_on_of_differentiable_on {π•œ : Type u_1} [nondiscrete_normed_field π•œ] {E : Type u_2} [normed_group E] [normed_space π•œ E] {F : Type u_3} [normed_group F] [normed_space π•œ F] {s : set E} {f : E β†’ F} {n : with_top β„•} :
(βˆ€ (m : β„•), ↑m ≀ n β†’ differentiable_on π•œ (iterated_fderiv_within π•œ m f s) s) β†’ times_cont_diff_on π•œ n f s

theorem times_cont_diff_on.continuous_on_iterated_fderiv_within {π•œ : Type u_1} [nondiscrete_normed_field π•œ] {E : Type u_2} [normed_group E] [normed_space π•œ E] {F : Type u_3} [normed_group F] [normed_space π•œ F] {s : set E} {f : E β†’ F} {n : with_top β„•} {m : β„•} :
times_cont_diff_on π•œ n f s β†’ ↑m ≀ n β†’ unique_diff_on π•œ s β†’ continuous_on (iterated_fderiv_within π•œ m f s) s

theorem times_cont_diff_on.differentiable_on_iterated_fderiv_within {π•œ : Type u_1} [nondiscrete_normed_field π•œ] {E : Type u_2} [normed_group E] [normed_space π•œ E] {F : Type u_3} [normed_group F] [normed_space π•œ F] {s : set E} {f : E β†’ F} {n : with_top β„•} {m : β„•} :
times_cont_diff_on π•œ n f s β†’ ↑m < n β†’ unique_diff_on π•œ s β†’ differentiable_on π•œ (iterated_fderiv_within π•œ m f s) s

theorem times_cont_diff_on_iff_continuous_on_differentiable_on {π•œ : Type u_1} [nondiscrete_normed_field π•œ] {E : Type u_2} [normed_group E] [normed_space π•œ E] {F : Type u_3} [normed_group F] [normed_space π•œ F] {s : set E} {f : E β†’ F} {n : with_top β„•} :
unique_diff_on π•œ s β†’ (times_cont_diff_on π•œ n f s ↔ (βˆ€ (m : β„•), ↑m ≀ n β†’ continuous_on (Ξ» (x : E), iterated_fderiv_within π•œ m f s x) s) ∧ βˆ€ (m : β„•), ↑m < n β†’ differentiable_on π•œ (Ξ» (x : E), iterated_fderiv_within π•œ m f s x) s)

theorem times_cont_diff_on_succ_iff_fderiv_within {π•œ : Type u_1} [nondiscrete_normed_field π•œ] {E : Type u_2} [normed_group E] [normed_space π•œ E] {F : Type u_3} [normed_group F] [normed_space π•œ F] {s : set E} {f : E β†’ F} {n : β„•} :
unique_diff_on π•œ s β†’ (times_cont_diff_on π•œ ↑(n + 1) f s ↔ differentiable_on π•œ f s ∧ times_cont_diff_on π•œ ↑n (Ξ» (y : E), fderiv_within π•œ f s y) s)

A function is C^(n + 1) on a domain with unique derivatives if and only if it is differentiable there, and its derivative (expressed with fderiv_within) is C^n.

theorem times_cont_diff_on_succ_iff_fderiv_of_open {π•œ : Type u_1} [nondiscrete_normed_field π•œ] {E : Type u_2} [normed_group E] [normed_space π•œ E] {F : Type u_3} [normed_group F] [normed_space π•œ F] {s : set E} {f : E β†’ F} {n : β„•} :
is_open s β†’ (times_cont_diff_on π•œ ↑(n + 1) f s ↔ differentiable_on π•œ f s ∧ times_cont_diff_on π•œ ↑n (Ξ» (y : E), fderiv π•œ f y) s)

A function is C^(n + 1) on an open domain if and only if it is differentiable there, and its derivative (expressed with fderiv) is C^n.

theorem times_cont_diff_on_top_iff_fderiv_within {π•œ : Type u_1} [nondiscrete_normed_field π•œ] {E : Type u_2} [normed_group E] [normed_space π•œ E] {F : Type u_3} [normed_group F] [normed_space π•œ F] {s : set E} {f : E β†’ F} :
unique_diff_on π•œ s β†’ (times_cont_diff_on π•œ ⊀ f s ↔ differentiable_on π•œ f s ∧ times_cont_diff_on π•œ ⊀ (Ξ» (y : E), fderiv_within π•œ f s y) s)

A function is C^∞ on a domain with unique derivatives if and only if it is differentiable there, and its derivative (expressed with fderiv_within) is C^∞.

theorem times_cont_diff_on_top_iff_fderiv_of_open {π•œ : Type u_1} [nondiscrete_normed_field π•œ] {E : Type u_2} [normed_group E] [normed_space π•œ E] {F : Type u_3} [normed_group F] [normed_space π•œ F] {s : set E} {f : E β†’ F} :
is_open s β†’ (times_cont_diff_on π•œ ⊀ f s ↔ differentiable_on π•œ f s ∧ times_cont_diff_on π•œ ⊀ (Ξ» (y : E), fderiv π•œ f y) s)

A function is C^∞ on an open domain if and only if it is differentiable there, and its derivative (expressed with fderiv) is C^∞.

theorem times_cont_diff_on.fderiv_within {π•œ : Type u_1} [nondiscrete_normed_field π•œ] {E : Type u_2} [normed_group E] [normed_space π•œ E] {F : Type u_3} [normed_group F] [normed_space π•œ F] {s : set E} {f : E β†’ F} {m n : with_top β„•} :
times_cont_diff_on π•œ n f s β†’ unique_diff_on π•œ s β†’ m + 1 ≀ n β†’ times_cont_diff_on π•œ m (Ξ» (y : E), fderiv_within π•œ f s y) s

theorem times_cont_diff_on.fderiv_of_open {π•œ : Type u_1} [nondiscrete_normed_field π•œ] {E : Type u_2} [normed_group E] [normed_space π•œ E] {F : Type u_3} [normed_group F] [normed_space π•œ F] {s : set E} {f : E β†’ F} {m n : with_top β„•} :
times_cont_diff_on π•œ n f s β†’ is_open s β†’ m + 1 ≀ n β†’ times_cont_diff_on π•œ m (Ξ» (y : E), fderiv π•œ f y) s

theorem times_cont_diff_on.continuous_on_fderiv_within {π•œ : Type u_1} [nondiscrete_normed_field π•œ] {E : Type u_2} [normed_group E] [normed_space π•œ E] {F : Type u_3} [normed_group F] [normed_space π•œ F] {s : set E} {f : E β†’ F} {n : with_top β„•} :
times_cont_diff_on π•œ n f s β†’ unique_diff_on π•œ s β†’ 1 ≀ n β†’ continuous_on (Ξ» (x : E), fderiv_within π•œ f s x) s

theorem times_cont_diff_on.continuous_on_fderiv_of_open {π•œ : Type u_1} [nondiscrete_normed_field π•œ] {E : Type u_2} [normed_group E] [normed_space π•œ E] {F : Type u_3} [normed_group F] [normed_space π•œ F] {s : set E} {f : E β†’ F} {n : with_top β„•} :
times_cont_diff_on π•œ n f s β†’ is_open s β†’ 1 ≀ n β†’ continuous_on (Ξ» (x : E), fderiv π•œ f x) s

theorem times_cont_diff_on.continuous_on_fderiv_within_apply {π•œ : Type u_1} [nondiscrete_normed_field π•œ] {E : Type u_2} [normed_group E] [normed_space π•œ E] {F : Type u_3} [normed_group F] [normed_space π•œ F] {s : set E} {f : E β†’ F} {n : with_top β„•} :
times_cont_diff_on π•œ n f s β†’ unique_diff_on π•œ s β†’ 1 ≀ n β†’ continuous_on (Ξ» (p : E Γ— E), ⇑(fderiv_within π•œ f s p.fst) p.snd) (s.prod set.univ)

If a function is at least C^1, its bundled derivative (mapping (x, v) to Df(x) v) is continuous.

Functions with a Taylor series on the whole space

structure has_ftaylor_series_up_to {π•œ : Type u_1} [nondiscrete_normed_field π•œ] {E : Type u_2} [normed_group E] [normed_space π•œ E] {F : Type u_3} [normed_group F] [normed_space π•œ F] :
with_top β„• β†’ (E β†’ F) β†’ (E β†’ formal_multilinear_series π•œ E F) β†’ Prop

has_ftaylor_series_up_to n f p registers the fact that p 0 = f and p (m+1) is a derivative of p m for m < n, and is continuous for m ≀ n. This is a predicate analogous to has_fderiv_at but for higher order derivatives.

theorem has_ftaylor_series_up_to.zero_eq' {π•œ : Type u_1} [nondiscrete_normed_field π•œ] {E : Type u_2} [normed_group E] [normed_space π•œ E] {F : Type u_3} [normed_group F] [normed_space π•œ F] {f : E β†’ F} {p : E β†’ formal_multilinear_series π•œ E F} {n : with_top β„•} (h : has_ftaylor_series_up_to n f p) (x : E) :
p x 0 = ⇑((continuous_multilinear_curry_fin0 π•œ E F).symm) (f x)

theorem has_ftaylor_series_up_to_on_univ_iff {π•œ : Type u_1} [nondiscrete_normed_field π•œ] {E : Type u_2} [normed_group E] [normed_space π•œ E] {F : Type u_3} [normed_group F] [normed_space π•œ F] {f : E β†’ F} {p : E β†’ formal_multilinear_series π•œ E F} {n : with_top β„•} :

theorem has_ftaylor_series_up_to.has_ftaylor_series_up_to_on {π•œ : Type u_1} [nondiscrete_normed_field π•œ] {E : Type u_2} [normed_group E] [normed_space π•œ E] {F : Type u_3} [normed_group F] [normed_space π•œ F] {f : E β†’ F} {p : E β†’ formal_multilinear_series π•œ E F} {n : with_top β„•} (h : has_ftaylor_series_up_to n f p) (s : set E) :

theorem has_ftaylor_series_up_to.of_le {π•œ : Type u_1} [nondiscrete_normed_field π•œ] {E : Type u_2} [normed_group E] [normed_space π•œ E] {F : Type u_3} [normed_group F] [normed_space π•œ F] {f : E β†’ F} {p : E β†’ formal_multilinear_series π•œ E F} {m n : with_top β„•} :

theorem has_ftaylor_series_up_to.continuous {π•œ : Type u_1} [nondiscrete_normed_field π•œ] {E : Type u_2} [normed_group E] [normed_space π•œ E] {F : Type u_3} [normed_group F] [normed_space π•œ F] {f : E β†’ F} {p : E β†’ formal_multilinear_series π•œ E F} {n : with_top β„•} :

theorem has_ftaylor_series_up_to_zero_iff {π•œ : Type u_1} [nondiscrete_normed_field π•œ] {E : Type u_2} [normed_group E] [normed_space π•œ E] {F : Type u_3} [normed_group F] [normed_space π•œ F] {f : E β†’ F} {p : E β†’ formal_multilinear_series π•œ E F} :
has_ftaylor_series_up_to 0 f p ↔ continuous f ∧ βˆ€ (x : E), (p x 0).uncurry0 = f x

theorem has_ftaylor_series_up_to.has_fderiv_at {π•œ : Type u_1} [nondiscrete_normed_field π•œ] {E : Type u_2} [normed_group E] [normed_space π•œ E] {F : Type u_3} [normed_group F] [normed_space π•œ F] {f : E β†’ F} {p : E β†’ formal_multilinear_series π•œ E F} {n : with_top β„•} (h : has_ftaylor_series_up_to n f p) (hn : 1 ≀ n) (x : E) :

If a function has a Taylor series at order at least 1, then the term of order 1 of this series is a derivative of f.

theorem has_ftaylor_series_up_to.differentiable {π•œ : Type u_1} [nondiscrete_normed_field π•œ] {E : Type u_2} [normed_group E] [normed_space π•œ E] {F : Type u_3} [normed_group F] [normed_space π•œ F] {f : E β†’ F} {p : E β†’ formal_multilinear_series π•œ E F} {n : with_top β„•} :
has_ftaylor_series_up_to n f p β†’ 1 ≀ n β†’ differentiable π•œ f

theorem has_ftaylor_series_up_to_succ_iff_right {π•œ : Type u_1} [nondiscrete_normed_field π•œ] {E : Type u_2} [normed_group E] [normed_space π•œ E] {F : Type u_3} [normed_group F] [normed_space π•œ F] {f : E β†’ F} {p : E β†’ formal_multilinear_series π•œ E F} {n : β„•} :
has_ftaylor_series_up_to ↑(n + 1) f p ↔ (βˆ€ (x : E), (p x 0).uncurry0 = f x) ∧ (βˆ€ (x : E), has_fderiv_at (Ξ» (y : E), p y 0) (p x 1).curry_left x) ∧ has_ftaylor_series_up_to ↑n (Ξ» (x : E), ⇑(continuous_multilinear_curry_fin1 π•œ E F) (p x 1)) (Ξ» (x : E), (p x).shift)

p is a Taylor series of f up to n+1 if and only if p.shift is a Taylor series up to n for p 1, which is a derivative of f.

Smooth functions at a point

def times_cont_diff_at (π•œ : Type u_1) [nondiscrete_normed_field π•œ] {E : Type u_2} [normed_group E] [normed_space π•œ E] {F : Type u_3} [normed_group F] [normed_space π•œ F] :
with_top β„• β†’ (E β†’ F) β†’ E β†’ Prop

A function is continuously differentiable up to n at a point x if, for any integer k ≀ n, there is a neighborhood of x where f admits derivatives up to order n, which are continuous.

Equations
theorem times_cont_diff_within_at_univ {π•œ : Type u_1} [nondiscrete_normed_field π•œ] {E : Type u_2} [normed_group E] [normed_space π•œ E] {F : Type u_3} [normed_group F] [normed_space π•œ F] {f : E β†’ F} {x : E} {n : with_top β„•} :

theorem times_cont_diff_at_top {π•œ : Type u_1} [nondiscrete_normed_field π•œ] {E : Type u_2} [normed_group E] [normed_space π•œ E] {F : Type u_3} [normed_group F] [normed_space π•œ F] {f : E β†’ F} {x : E} :
times_cont_diff_at π•œ ⊀ f x ↔ βˆ€ (n : β„•), times_cont_diff_at π•œ ↑n f x

theorem times_cont_diff_at.times_cont_diff_within_at {π•œ : Type u_1} [nondiscrete_normed_field π•œ] {E : Type u_2} [normed_group E] [normed_space π•œ E] {F : Type u_3} [normed_group F] [normed_space π•œ F] {s : set E} {f : E β†’ F} {x : E} {n : with_top β„•} :
times_cont_diff_at π•œ n f x β†’ times_cont_diff_within_at π•œ n f s x

theorem times_cont_diff_within_at.times_cont_diff_at {π•œ : Type u_1} [nondiscrete_normed_field π•œ] {E : Type u_2} [normed_group E] [normed_space π•œ E] {F : Type u_3} [normed_group F] [normed_space π•œ F] {s : set E} {f : E β†’ F} {x : E} {n : with_top β„•} :
times_cont_diff_within_at π•œ n f s x β†’ s ∈ 𝓝 x β†’ times_cont_diff_at π•œ n f x

theorem times_cont_diff_at.of_le {π•œ : Type u_1} [nondiscrete_normed_field π•œ] {E : Type u_2} [normed_group E] [normed_space π•œ E] {F : Type u_3} [normed_group F] [normed_space π•œ F] {f : E β†’ F} {x : E} {m n : with_top β„•} :
times_cont_diff_at π•œ n f x β†’ m ≀ n β†’ times_cont_diff_at π•œ m f x

theorem times_cont_diff_at.continuous_at {π•œ : Type u_1} [nondiscrete_normed_field π•œ] {E : Type u_2} [normed_group E] [normed_space π•œ E] {F : Type u_3} [normed_group F] [normed_space π•œ F] {f : E β†’ F} {x : E} {n : with_top β„•} :
times_cont_diff_at π•œ n f x β†’ continuous_at f x

theorem times_cont_diff_at.differentiable {π•œ : Type u_1} [nondiscrete_normed_field π•œ] {E : Type u_2} [normed_group E] [normed_space π•œ E] {F : Type u_3} [normed_group F] [normed_space π•œ F] {f : E β†’ F} {x : E} {n : with_top β„•} :
times_cont_diff_at π•œ n f x β†’ 1 ≀ n β†’ differentiable_at π•œ f x

If a function is C^n with n β‰₯ 1 at a point, then it is differentiable there.

theorem times_cont_diff_at_succ_iff_has_fderiv_at {π•œ : Type u_1} [nondiscrete_normed_field π•œ] {E : Type u_2} [normed_group E] [normed_space π•œ E] {F : Type u_3} [normed_group F] [normed_space π•œ F] {f : E β†’ F} {x : E} {n : β„•} :
times_cont_diff_at π•œ ↑(n + 1) f x ↔ βˆƒ (f' : E β†’ (E β†’L[π•œ] F)), (βˆƒ (u : set E) (H : u ∈ 𝓝 x), βˆ€ (x : E), x ∈ u β†’ has_fderiv_at f (f' x) x) ∧ times_cont_diff_at π•œ ↑n f' x

A function is C^(n + 1) at a point iff locally, it has a derivative which is C^n.

Smooth functions

def times_cont_diff (π•œ : Type u_1) [nondiscrete_normed_field π•œ] {E : Type u_2} [normed_group E] [normed_space π•œ E] {F : Type u_3} [normed_group F] [normed_space π•œ F] :
with_top β„• β†’ (E β†’ F) β†’ Prop

A function is continuously differentiable up to n if it admits derivatives up to order n, which are continuous. Contrary to the case of definitions in domains (where derivatives might not be unique) we do not need to localize the definition in space or time.

Equations
theorem times_cont_diff_on_univ {π•œ : Type u_1} [nondiscrete_normed_field π•œ] {E : Type u_2} [normed_group E] [normed_space π•œ E] {F : Type u_3} [normed_group F] [normed_space π•œ F] {f : E β†’ F} {n : with_top β„•} :

theorem times_cont_diff_iff_times_cont_diff_at {π•œ : Type u_1} [nondiscrete_normed_field π•œ] {E : Type u_2} [normed_group E] [normed_space π•œ E] {F : Type u_3} [normed_group F] [normed_space π•œ F] {f : E β†’ F} {n : with_top β„•} :
times_cont_diff π•œ n f ↔ βˆ€ (x : E), times_cont_diff_at π•œ n f x

theorem times_cont_diff.times_cont_diff_at {π•œ : Type u_1} [nondiscrete_normed_field π•œ] {E : Type u_2} [normed_group E] [normed_space π•œ E] {F : Type u_3} [normed_group F] [normed_space π•œ F] {f : E β†’ F} {x : E} {n : with_top β„•} :
times_cont_diff π•œ n f β†’ times_cont_diff_at π•œ n f x

theorem times_cont_diff.times_cont_diff_within_at {π•œ : Type u_1} [nondiscrete_normed_field π•œ] {E : Type u_2} [normed_group E] [normed_space π•œ E] {F : Type u_3} [normed_group F] [normed_space π•œ F] {s : set E} {f : E β†’ F} {x : E} {n : with_top β„•} :
times_cont_diff π•œ n f β†’ times_cont_diff_within_at π•œ n f s x

theorem times_cont_diff_top {π•œ : Type u_1} [nondiscrete_normed_field π•œ] {E : Type u_2} [normed_group E] [normed_space π•œ E] {F : Type u_3} [normed_group F] [normed_space π•œ F] {f : E β†’ F} :
times_cont_diff π•œ ⊀ f ↔ βˆ€ (n : β„•), times_cont_diff π•œ ↑n f

theorem times_cont_diff.times_cont_diff_on {π•œ : Type u_1} [nondiscrete_normed_field π•œ] {E : Type u_2} [normed_group E] [normed_space π•œ E] {F : Type u_3} [normed_group F] [normed_space π•œ F] {s : set E} {f : E β†’ F} {n : with_top β„•} :
times_cont_diff π•œ n f β†’ times_cont_diff_on π•œ n f s

@[simp]
theorem times_cont_diff_zero {π•œ : Type u_1} [nondiscrete_normed_field π•œ] {E : Type u_2} [normed_group E] [normed_space π•œ E] {F : Type u_3} [normed_group F] [normed_space π•œ F] {f : E β†’ F} :

theorem times_cont_diff_at_zero {π•œ : Type u_1} [nondiscrete_normed_field π•œ] {E : Type u_2} [normed_group E] [normed_space π•œ E] {F : Type u_3} [normed_group F] [normed_space π•œ F] {f : E β†’ F} {x : E} :
times_cont_diff_at π•œ 0 f x ↔ βˆƒ (u : set E) (H : u ∈ 𝓝 x), continuous_on f u

theorem times_cont_diff.of_le {π•œ : Type u_1} [nondiscrete_normed_field π•œ] {E : Type u_2} [normed_group E] [normed_space π•œ E] {F : Type u_3} [normed_group F] [normed_space π•œ F] {f : E β†’ F} {m n : with_top β„•} :
times_cont_diff π•œ n f β†’ m ≀ n β†’ times_cont_diff π•œ m f

theorem times_cont_diff.continuous {π•œ : Type u_1} [nondiscrete_normed_field π•œ] {E : Type u_2} [normed_group E] [normed_space π•œ E] {F : Type u_3} [normed_group F] [normed_space π•œ F] {f : E β†’ F} {n : with_top β„•} :
times_cont_diff π•œ n f β†’ continuous f

theorem times_cont_diff.differentiable {π•œ : Type u_1} [nondiscrete_normed_field π•œ] {E : Type u_2} [normed_group E] [normed_space π•œ E] {F : Type u_3} [normed_group F] [normed_space π•œ F] {f : E β†’ F} {n : with_top β„•} :
times_cont_diff π•œ n f β†’ 1 ≀ n β†’ differentiable π•œ f

If a function is C^n with n β‰₯ 1, then it is differentiable.

Iterated derivative

def iterated_fderiv (π•œ : Type u_1) [nondiscrete_normed_field π•œ] {E : Type u_2} [normed_group E] [normed_space π•œ E] {F : Type u_3} [normed_group F] [normed_space π•œ F] (n : β„•) :
(E β†’ F) β†’ E β†’ continuous_multilinear_map π•œ (Ξ» (i : fin n), E) F

The n-th derivative of a function, as a multilinear map, defined inductively.

Equations
def ftaylor_series (π•œ : Type u_1) [nondiscrete_normed_field π•œ] {E : Type u_2} [normed_group E] [normed_space π•œ E] {F : Type u_3} [normed_group F] [normed_space π•œ F] :
(E β†’ F) β†’ E β†’ formal_multilinear_series π•œ E F

Formal Taylor series associated to a function within a set.

Equations
@[simp]
theorem iterated_fderiv_zero_apply {π•œ : Type u_1} [nondiscrete_normed_field π•œ] {E : Type u_2} [normed_group E] [normed_space π•œ E] {F : Type u_3} [normed_group F] [normed_space π•œ F] {f : E β†’ F} {x : E} (m : fin 0 β†’ E) :
⇑(iterated_fderiv π•œ 0 f x) m = f x

theorem iterated_fderiv_zero_eq_comp {π•œ : Type u_1} [nondiscrete_normed_field π•œ] {E : Type u_2} [normed_group E] [normed_space π•œ E] {F : Type u_3} [normed_group F] [normed_space π•œ F] {f : E β†’ F} :

theorem iterated_fderiv_succ_apply_left {π•œ : Type u_1} [nondiscrete_normed_field π•œ] {E : Type u_2} [normed_group E] [normed_space π•œ E] {F : Type u_3} [normed_group F] [normed_space π•œ F] {f : E β†’ F} {x : E} {n : β„•} (m : fin (n + 1) β†’ E) :
⇑(iterated_fderiv π•œ (n + 1) f x) m = ⇑(⇑(fderiv π•œ (iterated_fderiv π•œ n f) x) (m 0)) (fin.tail m)

theorem iterated_fderiv_succ_eq_comp_left {π•œ : Type u_1} [nondiscrete_normed_field π•œ] {E : Type u_2} [normed_group E] [normed_space π•œ E] {F : Type u_3} [normed_group F] [normed_space π•œ F] {f : E β†’ F} {n : β„•} :
iterated_fderiv π•œ (n + 1) f = ⇑(continuous_multilinear_curry_left_equiv π•œ (Ξ» (i : fin (n + 1)), E) F) ∘ fderiv π•œ (iterated_fderiv π•œ n f)

Writing explicitly the n+1-th derivative as the composition of a currying linear equiv, and the derivative of the n-th derivative.

theorem iterated_fderiv_within_univ {π•œ : Type u_1} [nondiscrete_normed_field π•œ] {E : Type u_2} [normed_group E] [normed_space π•œ E] {F : Type u_3} [normed_group F] [normed_space π•œ F] {f : E β†’ F} {n : β„•} :

theorem ftaylor_series_within_univ {π•œ : Type u_1} [nondiscrete_normed_field π•œ] {E : Type u_2} [normed_group E] [normed_space π•œ E] {F : Type u_3} [normed_group F] [normed_space π•œ F] {f : E β†’ F} :

theorem iterated_fderiv_succ_apply_right {π•œ : Type u_1} [nondiscrete_normed_field π•œ] {E : Type u_2} [normed_group E] [normed_space π•œ E] {F : Type u_3} [normed_group F] [normed_space π•œ F] {f : E β†’ F} {x : E} {n : β„•} (m : fin (n + 1) β†’ E) :
⇑(iterated_fderiv π•œ (n + 1) f x) m = ⇑(⇑(iterated_fderiv π•œ n (Ξ» (y : E), fderiv π•œ f y) x) (fin.init m)) (m (fin.last n))

theorem iterated_fderiv_succ_eq_comp_right {π•œ : Type u_1} [nondiscrete_normed_field π•œ] {E : Type u_2} [normed_group E] [normed_space π•œ E] {F : Type u_3} [normed_group F] [normed_space π•œ F] {f : E β†’ F} {x : E} {n : β„•} :
iterated_fderiv π•œ (n + 1) f x = (⇑(continuous_multilinear_curry_right_equiv π•œ (Ξ» (i : fin (n + 1)), E) F) ∘ iterated_fderiv π•œ n (Ξ» (y : E), fderiv π•œ f y)) x

Writing explicitly the n+1-th derivative as the composition of a currying linear equiv, and the n-th derivative of the derivative.

@[simp]
theorem iterated_fderiv_one_apply {π•œ : Type u_1} [nondiscrete_normed_field π•œ] {E : Type u_2} [normed_group E] [normed_space π•œ E] {F : Type u_3} [normed_group F] [normed_space π•œ F] {f : E β†’ F} {x : E} (m : fin 1 β†’ E) :
⇑(iterated_fderiv π•œ 1 f x) m = ⇑(fderiv π•œ f x) (m 0)

theorem times_cont_diff_on_iff_ftaylor_series {π•œ : Type u_1} [nondiscrete_normed_field π•œ] {E : Type u_2} [normed_group E] [normed_space π•œ E] {F : Type u_3} [normed_group F] [normed_space π•œ F] {f : E β†’ F} {n : with_top β„•} :

When a function is C^n in a set s of unique differentiability, it admits ftaylor_series_within π•œ f s as a Taylor series up to order n in s.

theorem times_cont_diff_iff_continuous_differentiable {π•œ : Type u_1} [nondiscrete_normed_field π•œ] {E : Type u_2} [normed_group E] [normed_space π•œ E] {F : Type u_3} [normed_group F] [normed_space π•œ F] {f : E β†’ F} {n : with_top β„•} :
times_cont_diff π•œ n f ↔ (βˆ€ (m : β„•), ↑m ≀ n β†’ continuous (Ξ» (x : E), iterated_fderiv π•œ m f x)) ∧ βˆ€ (m : β„•), ↑m < n β†’ differentiable π•œ (Ξ» (x : E), iterated_fderiv π•œ m f x)

theorem times_cont_diff_of_differentiable_iterated_fderiv {π•œ : Type u_1} [nondiscrete_normed_field π•œ] {E : Type u_2} [normed_group E] [normed_space π•œ E] {F : Type u_3} [normed_group F] [normed_space π•œ F] {f : E β†’ F} {n : with_top β„•} :
(βˆ€ (m : β„•), ↑m ≀ n β†’ differentiable π•œ (iterated_fderiv π•œ m f)) β†’ times_cont_diff π•œ n f

theorem times_cont_diff_succ_iff_fderiv {π•œ : Type u_1} [nondiscrete_normed_field π•œ] {E : Type u_2} [normed_group E] [normed_space π•œ E] {F : Type u_3} [normed_group F] [normed_space π•œ F] {f : E β†’ F} {n : β„•} :
times_cont_diff π•œ ↑(n + 1) f ↔ differentiable π•œ f ∧ times_cont_diff π•œ ↑n (Ξ» (y : E), fderiv π•œ f y)

A function is C^(n + 1) on a domain with unique derivatives if and only if it is differentiable there, and its derivative is C^n.

theorem times_cont_diff_top_iff_fderiv {π•œ : Type u_1} [nondiscrete_normed_field π•œ] {E : Type u_2} [normed_group E] [normed_space π•œ E] {F : Type u_3} [normed_group F] [normed_space π•œ F] {f : E β†’ F} :
times_cont_diff π•œ ⊀ f ↔ differentiable π•œ f ∧ times_cont_diff π•œ ⊀ (Ξ» (y : E), fderiv π•œ f y)

A function is C^∞ on a domain with unique derivatives if and only if it is differentiable there, and its derivative is C^∞.

theorem times_cont_diff.continuous_fderiv {π•œ : Type u_1} [nondiscrete_normed_field π•œ] {E : Type u_2} [normed_group E] [normed_space π•œ E] {F : Type u_3} [normed_group F] [normed_space π•œ F] {f : E β†’ F} {n : with_top β„•} :
times_cont_diff π•œ n f β†’ 1 ≀ n β†’ continuous (Ξ» (x : E), fderiv π•œ f x)

theorem times_cont_diff.continuous_fderiv_apply {π•œ : Type u_1} [nondiscrete_normed_field π•œ] {E : Type u_2} [normed_group E] [normed_space π•œ E] {F : Type u_3} [normed_group F] [normed_space π•œ F] {f : E β†’ F} {n : with_top β„•} :
times_cont_diff π•œ n f β†’ 1 ≀ n β†’ continuous (Ξ» (p : E Γ— E), ⇑(fderiv π•œ f p.fst) p.snd)

If a function is at least C^1, its bundled derivative (mapping (x, v) to Df(x) v) is continuous.

Constants

theorem iterated_fderiv_within_zero_fun {π•œ : Type u_1} [nondiscrete_normed_field π•œ] {E : Type u_2} [normed_group E] [normed_space π•œ E] {F : Type u_3} [normed_group F] [normed_space π•œ F] {n : β„•} :
iterated_fderiv π•œ n (Ξ» (x : E), 0) = 0

theorem times_cont_diff_zero_fun {π•œ : Type u_1} [nondiscrete_normed_field π•œ] {E : Type u_2} [normed_group E] [normed_space π•œ E] {F : Type u_3} [normed_group F] [normed_space π•œ F] {n : with_top β„•} :
times_cont_diff π•œ n (Ξ» (x : E), 0)

theorem times_cont_diff_const {π•œ : Type u_1} [nondiscrete_normed_field π•œ] {E : Type u_2} [normed_group E] [normed_space π•œ E] {F : Type u_3} [normed_group F] [normed_space π•œ F] {n : with_top β„•} {c : F} :
times_cont_diff π•œ n (Ξ» (x : E), c)

Constants are C^∞.

theorem times_cont_diff_on_const {π•œ : Type u_1} [nondiscrete_normed_field π•œ] {E : Type u_2} [normed_group E] [normed_space π•œ E] {F : Type u_3} [normed_group F] [normed_space π•œ F] {n : with_top β„•} {c : F} {s : set E} :
times_cont_diff_on π•œ n (Ξ» (x : E), c) s

theorem times_cont_diff_at_const {π•œ : Type u_1} [nondiscrete_normed_field π•œ] {E : Type u_2} [normed_group E] [normed_space π•œ E] {F : Type u_3} [normed_group F] [normed_space π•œ F] {x : E} {n : with_top β„•} {c : F} :
times_cont_diff_at π•œ n (Ξ» (x : E), c) x

theorem times_cont_diff_within_at_const {π•œ : Type u_1} [nondiscrete_normed_field π•œ] {E : Type u_2} [normed_group E] [normed_space π•œ E] {F : Type u_3} [normed_group F] [normed_space π•œ F] {s : set E} {x : E} {n : with_top β„•} {c : F} :
times_cont_diff_within_at π•œ n (Ξ» (x : E), c) s x

theorem times_cont_diff_of_subsingleton {π•œ : Type u_1} [nondiscrete_normed_field π•œ] {E : Type u_2} [normed_group E] [normed_space π•œ E] {F : Type u_3} [normed_group F] [normed_space π•œ F] {f : E β†’ F} [subsingleton F] {n : with_top β„•} :
times_cont_diff π•œ n f

theorem times_cont_diff_at_of_subsingleton {π•œ : Type u_1} [nondiscrete_normed_field π•œ] {E : Type u_2} [normed_group E] [normed_space π•œ E] {F : Type u_3} [normed_group F] [normed_space π•œ F] {f : E β†’ F} {x : E} [subsingleton F] {n : with_top β„•} :
times_cont_diff_at π•œ n f x

theorem times_cont_diff_within_at_of_subsingleton {π•œ : Type u_1} [nondiscrete_normed_field π•œ] {E : Type u_2} [normed_group E] [normed_space π•œ E] {F : Type u_3} [normed_group F] [normed_space π•œ F] {s : set E} {f : E β†’ F} {x : E} [subsingleton F] {n : with_top β„•} :
times_cont_diff_within_at π•œ n f s x

theorem times_cont_diff_on_of_subsingleton {π•œ : Type u_1} [nondiscrete_normed_field π•œ] {E : Type u_2} [normed_group E] [normed_space π•œ E] {F : Type u_3} [normed_group F] [normed_space π•œ F] {s : set E} {f : E β†’ F} [subsingleton F] {n : with_top β„•} :
times_cont_diff_on π•œ n f s

Linear functions

theorem is_bounded_linear_map.times_cont_diff {π•œ : Type u_1} [nondiscrete_normed_field π•œ] {E : Type u_2} [normed_group E] [normed_space π•œ E] {F : Type u_3} [normed_group F] [normed_space π•œ F] {f : E β†’ F} {n : with_top β„•} :
is_bounded_linear_map π•œ f β†’ times_cont_diff π•œ n f

Unbundled bounded linear functions are C^∞.

theorem continuous_linear_map.times_cont_diff {π•œ : Type u_1} [nondiscrete_normed_field π•œ] {E : Type u_2} [normed_group E] [normed_space π•œ E] {F : Type u_3} [normed_group F] [normed_space π•œ F] {n : with_top β„•} (f : E β†’L[π•œ] F) :

theorem times_cont_diff_fst {π•œ : Type u_1} [nondiscrete_normed_field π•œ] {E : Type u_2} [normed_group E] [normed_space π•œ E] {F : Type u_3} [normed_group F] [normed_space π•œ F] {n : with_top β„•} :

The first projection in a product is C^∞.

theorem times_cont_diff_on_fst {π•œ : Type u_1} [nondiscrete_normed_field π•œ] {E : Type u_2} [normed_group E] [normed_space π•œ E] {F : Type u_3} [normed_group F] [normed_space π•œ F] {s : set (E Γ— F)} {n : with_top β„•} :

The first projection on a domain in a product is C^∞.

theorem times_cont_diff_at_fst {π•œ : Type u_1} [nondiscrete_normed_field π•œ] {E : Type u_2} [normed_group E] [normed_space π•œ E] {F : Type u_3} [normed_group F] [normed_space π•œ F] {p : E Γ— F} {n : with_top β„•} :

The first projection at a point in a product is C^∞.

theorem times_cont_diff_within_at_fst {π•œ : Type u_1} [nondiscrete_normed_field π•œ] {E : Type u_2} [normed_group E] [normed_space π•œ E] {F : Type u_3} [normed_group F] [normed_space π•œ F] {s : set (E Γ— F)} {p : E Γ— F} {n : with_top β„•} :

The first projection within a domain at a point in a product is C^∞.

theorem times_cont_diff_snd {π•œ : Type u_1} [nondiscrete_normed_field π•œ] {E : Type u_2} [normed_group E] [normed_space π•œ E] {F : Type u_3} [normed_group F] [normed_space π•œ F] {n : with_top β„•} :

The second projection in a product is C^∞.

theorem times_cont_diff_on_snd {π•œ : Type u_1} [nondiscrete_normed_field π•œ] {E : Type u_2} [normed_group E] [normed_space π•œ E] {F : Type u_3} [normed_group F] [normed_space π•œ F] {s : set (E Γ— F)} {n : with_top β„•} :

The second projection on a domain in a product is C^∞.

theorem times_cont_diff_at_snd {π•œ : Type u_1} [nondiscrete_normed_field π•œ] {E : Type u_2} [normed_group E] [normed_space π•œ E] {F : Type u_3} [normed_group F] [normed_space π•œ F] {p : E Γ— F} {n : with_top β„•} :

The second projection at a point in a product is C^∞.

theorem times_cont_diff_within_at_snd {π•œ : Type u_1} [nondiscrete_normed_field π•œ] {E : Type u_2} [normed_group E] [normed_space π•œ E] {F : Type u_3} [normed_group F] [normed_space π•œ F] {s : set (E Γ— F)} {p : E Γ— F} {n : with_top β„•} :

The second projection within a domain at a point in a product is C^∞.

theorem times_cont_diff_id {π•œ : Type u_1} [nondiscrete_normed_field π•œ] {E : Type u_2} [normed_group E] [normed_space π•œ E] {n : with_top β„•} :
times_cont_diff π•œ n id

The identity is C^∞.

theorem is_bounded_bilinear_map.times_cont_diff {π•œ : Type u_1} [nondiscrete_normed_field π•œ] {E : Type u_2} [normed_group E] [normed_space π•œ E] {F : Type u_3} [normed_group F] [normed_space π•œ F] {G : Type u_4} [normed_group G] [normed_space π•œ G] {b : E Γ— F β†’ G} {n : with_top β„•} :
is_bounded_bilinear_map π•œ b β†’ times_cont_diff π•œ n b

Bilinear functions are C^∞.

theorem has_ftaylor_series_up_to_on.continuous_linear_map_comp {π•œ : Type u_1} [nondiscrete_normed_field π•œ] {E : Type u_2} [normed_group E] [normed_space π•œ E] {F : Type u_3} [normed_group F] [normed_space π•œ F] {G : Type u_4} [normed_group G] [normed_space π•œ G] {s : set E} {f : E β†’ F} {p : E β†’ formal_multilinear_series π•œ E F} {n : with_top β„•} (g : F β†’L[π•œ] G) :

If f admits a Taylor series p in a set s, and g is linear, then g ∘ f admits a Taylor series whose k-th term is given by g ∘ (p k).

theorem times_cont_diff_within_at.continuous_linear_map_comp {π•œ : Type u_1} [nondiscrete_normed_field π•œ] {E : Type u_2} [normed_group E] [normed_space π•œ E] {F : Type u_3} [normed_group F] [normed_space π•œ F] {G : Type u_4} [normed_group G] [normed_space π•œ G] {s : set E} {f : E β†’ F} {x : E} {n : with_top β„•} (g : F β†’L[π•œ] G) :
times_cont_diff_within_at π•œ n f s x β†’ times_cont_diff_within_at π•œ n (⇑g ∘ f) s x

Composition by continuous linear maps on the left preserves C^n functions in a domain at a point.

theorem times_cont_diff_at.continuous_linear_map_comp {π•œ : Type u_1} [nondiscrete_normed_field π•œ] {E : Type u_2} [normed_group E] [normed_space π•œ E] {F : Type u_3} [normed_group F] [normed_space π•œ F] {G : Type u_4} [normed_group G] [normed_space π•œ G] {f : E β†’ F} {x : E} {n : with_top β„•} (g : F β†’L[π•œ] G) :
times_cont_diff_at π•œ n f x β†’ times_cont_diff_at π•œ n (⇑g ∘ f) x

Composition by continuous linear maps on the left preserves C^n functions in a domain at a point.

theorem times_cont_diff_on.continuous_linear_map_comp {π•œ : Type u_1} [nondiscrete_normed_field π•œ] {E : Type u_2} [normed_group E] [normed_space π•œ E] {F : Type u_3} [normed_group F] [normed_space π•œ F] {G : Type u_4} [normed_group G] [normed_space π•œ G] {s : set E} {f : E β†’ F} {n : with_top β„•} (g : F β†’L[π•œ] G) :
times_cont_diff_on π•œ n f s β†’ times_cont_diff_on π•œ n (⇑g ∘ f) s

Composition by continuous linear maps on the left preserves C^n functions on domains.

theorem times_cont_diff.continuous_linear_map_comp {π•œ : Type u_1} [nondiscrete_normed_field π•œ] {E : Type u_2} [normed_group E] [normed_space π•œ E] {F : Type u_3} [normed_group F] [normed_space π•œ F] {G : Type u_4} [normed_group G] [normed_space π•œ G] {n : with_top β„•} {f : E β†’ F} (g : F β†’L[π•œ] G) :
times_cont_diff π•œ n f β†’ times_cont_diff π•œ n (Ξ» (x : E), ⇑g (f x))

Composition by continuous linear maps on the left preserves C^n functions.

theorem continuous_linear_equiv.comp_times_cont_diff_within_at_iff {π•œ : Type u_1} [nondiscrete_normed_field π•œ] {E : Type u_2} [normed_group E] [normed_space π•œ E] {F : Type u_3} [normed_group F] [normed_space π•œ F] {G : Type u_4} [normed_group G] [normed_space π•œ G] {s : set E} {f : E β†’ F} {x : E} {n : with_top β„•} (e : F ≃L[π•œ] G) :

Composition by continuous linear equivs on the left respects higher differentiability on domains.

theorem continuous_linear_equiv.comp_times_cont_diff_on_iff {π•œ : Type u_1} [nondiscrete_normed_field π•œ] {E : Type u_2} [normed_group E] [normed_space π•œ E] {F : Type u_3} [normed_group F] [normed_space π•œ F] {G : Type u_4} [normed_group G] [normed_space π•œ G] {s : set E} {f : E β†’ F} {n : with_top β„•} (e : F ≃L[π•œ] G) :
times_cont_diff_on π•œ n (⇑e ∘ f) s ↔ times_cont_diff_on π•œ n f s

Composition by continuous linear equivs on the left respects higher differentiability on domains.

theorem has_ftaylor_series_up_to_on.comp_continuous_linear_map {π•œ : Type u_1} [nondiscrete_normed_field π•œ] {E : Type u_2} [normed_group E] [normed_space π•œ E] {F : Type u_3} [normed_group F] [normed_space π•œ F] {G : Type u_4} [normed_group G] [normed_space π•œ G] {s : set E} {f : E β†’ F} {p : E β†’ formal_multilinear_series π•œ E F} {n : with_top β„•} (hf : has_ftaylor_series_up_to_on n f p s) (g : G β†’L[π•œ] E) :
has_ftaylor_series_up_to_on n (f ∘ ⇑g) (Ξ» (x : G) (k : β„•), (p (⇑g x) k).comp_continuous_linear_map (Ξ» (_x : fin k), g)) (⇑g ⁻¹' s)

If f admits a Taylor series p in a set s, and g is linear, then f ∘ g admits a Taylor series in g ⁻¹' s, whose k-th term is given by p k (g v₁, ..., g vβ‚–) .

theorem times_cont_diff_within_at.comp_continuous_linear_map {π•œ : Type u_1} [nondiscrete_normed_field π•œ] {E : Type u_2} [normed_group E] [normed_space π•œ E] {F : Type u_3} [normed_group F] [normed_space π•œ F] {G : Type u_4} [normed_group G] [normed_space π•œ G] {s : set E} {f : E β†’ F} {n : with_top β„•} {x : G} (g : G β†’L[π•œ] E) :

Composition by continuous linear maps on the right preserves C^n functions at a point on a domain.

theorem times_cont_diff_on.comp_continuous_linear_map {π•œ : Type u_1} [nondiscrete_normed_field π•œ] {E : Type u_2} [normed_group E] [normed_space π•œ E] {F : Type u_3} [normed_group F] [normed_space π•œ F] {G : Type u_4} [normed_group G] [normed_space π•œ G] {s : set E} {f : E β†’ F} {n : with_top β„•} (hf : times_cont_diff_on π•œ n f s) (g : G β†’L[π•œ] E) :

Composition by continuous linear maps on the right preserves C^n functions on domains.

theorem times_cont_diff.comp_continuous_linear_map {π•œ : Type u_1} [nondiscrete_normed_field π•œ] {E : Type u_2} [normed_group E] [normed_space π•œ E] {F : Type u_3} [normed_group F] [normed_space π•œ F] {G : Type u_4} [normed_group G] [normed_space π•œ G] {n : with_top β„•} {f : E β†’ F} {g : G β†’L[π•œ] E} :
times_cont_diff π•œ n f β†’ times_cont_diff π•œ n (f ∘ ⇑g)

Composition by continuous linear maps on the right preserves C^n functions.

theorem continuous_linear_equiv.times_cont_diff_within_at_comp_iff {π•œ : Type u_1} [nondiscrete_normed_field π•œ] {E : Type u_2} [normed_group E] [normed_space π•œ E] {F : Type u_3} [normed_group F] [normed_space π•œ F] {G : Type u_4} [normed_group G] [normed_space π•œ G] {s : set E} {f : E β†’ F} {x : E} {n : with_top β„•} (e : G ≃L[π•œ] E) :

Composition by continuous linear equivs on the right respects higher differentiability at a point in a domain.

theorem continuous_linear_equiv.times_cont_diff_on_comp_iff {π•œ : Type u_1} [nondiscrete_normed_field π•œ] {E : Type u_2} [normed_group E] [normed_space π•œ E] {F : Type u_3} [normed_group F] [normed_space π•œ F] {G : Type u_4} [normed_group G] [normed_space π•œ G] {s : set E} {f : E β†’ F} {n : with_top β„•} (e : G ≃L[π•œ] E) :

Composition by continuous linear equivs on the right respects higher differentiability on domains.

theorem has_ftaylor_series_up_to_on.prod {π•œ : Type u_1} [nondiscrete_normed_field π•œ] {E : Type u_2} [normed_group E] [normed_space π•œ E] {F : Type u_3} [normed_group F] [normed_space π•œ F] {G : Type u_4} [normed_group G] [normed_space π•œ G] {s : set E} {f : E β†’ F} {p : E β†’ formal_multilinear_series π•œ E F} {n : with_top β„•} (hf : has_ftaylor_series_up_to_on n f p s) {g : E β†’ G} {q : E β†’ formal_multilinear_series π•œ E G} :
has_ftaylor_series_up_to_on n g q s β†’ has_ftaylor_series_up_to_on n (Ξ» (y : E), (f y, g y)) (Ξ» (y : E) (k : β„•), (p y k).prod (q y k)) s

If two functions f and g admit Taylor series p and q in a set s, then the cartesian product of f and g admits the cartesian product of p and q as a Taylor series.

theorem times_cont_diff_within_at.prod {π•œ : Type u_1} [nondiscrete_normed_field π•œ] {E : Type u_2} [normed_group E] [normed_space π•œ E] {F : Type u_3} [normed_group F] [normed_space π•œ F] {G : Type u_4} [normed_group G] [normed_space π•œ G] {x : E} {n : with_top β„•} {s : set E} {f : E β†’ F} {g : E β†’ G} :
times_cont_diff_within_at π•œ n f s x β†’ times_cont_diff_within_at π•œ n g s x β†’ times_cont_diff_within_at π•œ n (Ξ» (x : E), (f x, g x)) s x

The cartesian product of C^n functions at a point in a domain is C^n.

theorem times_cont_diff_on.prod {π•œ : Type u_1} [nondiscrete_normed_field π•œ] {E : Type u_2} [normed_group E] [normed_space π•œ E] {F : Type u_3} [normed_group F] [normed_space π•œ F] {G : Type u_4} [normed_group G] [normed_space π•œ G] {n : with_top β„•} {s : set E} {f : E β†’ F} {g : E β†’ G} :
times_cont_diff_on π•œ n f s β†’ times_cont_diff_on π•œ n g s β†’ times_cont_diff_on π•œ n (Ξ» (x : E), (f x, g x)) s

The cartesian product of C^n functions on domains is C^n.

theorem times_cont_diff_at.prod {π•œ : Type u_1} [nondiscrete_normed_field π•œ] {E : Type u_2} [normed_group E] [normed_space π•œ E] {F : Type u_3} [normed_group F] [normed_space π•œ F] {G : Type u_4} [normed_group G] [normed_space π•œ G] {x : E} {n : with_top β„•} {f : E β†’ F} {g : E β†’ G} :
times_cont_diff_at π•œ n f x β†’ times_cont_diff_at π•œ n g x β†’ times_cont_diff_at π•œ n (Ξ» (x : E), (f x, g x)) x

The cartesian product of C^n functions at a point is C^n.

theorem times_cont_diff.prod {π•œ : Type u_1} [nondiscrete_normed_field π•œ] {E : Type u_2} [normed_group E] [normed_space π•œ E] {F : Type u_3} [normed_group F] [normed_space π•œ F] {G : Type u_4} [normed_group G] [normed_space π•œ G] {n : with_top β„•} {f : E β†’ F} {g : E β†’ G} :
times_cont_diff π•œ n f β†’ times_cont_diff π•œ n g β†’ times_cont_diff π•œ n (Ξ» (x : E), (f x, g x))

The cartesian product of C^n functions is C^n.

Composition of C^n functions

We show that the composition of C^n functions is C^n. One way to prove it would be to write the n-th derivative of the composition (this is FaΓ  di Bruno's formula) and check its continuity, but this is very painful. Instead, we go for a simple inductive proof. Assume it is done for n. Then, to check it for n+1, one needs to check that the derivative of g ∘ f is C^n, i.e., that Dg(f x) ⬝ Df(x) is C^n. The term Dg (f x) is the composition of two C^n functions, so it is C^n by the inductive assumption. The term Df(x) is also C^n. Then, the matrix multiplication is the application of a bilinear map (which is C^∞, and therefore C^n) to x ↦ (Dg(f x), Df x). As the composition of two C^n maps, it is again C^n, and we are done.

There is a subtlety in this argument: we apply the inductive assumption to functions on other Banach spaces. In maths, one would say: prove by induction over n that, for all C^n maps between all pairs of Banach spaces, their composition is C^n. In Lean, this is fine as long as the spaces stay in the same universe. This is not the case in the above argument: if E lives in universe u and F lives in universe v, then linear maps from E to F (to which the derivative of f belongs) is in universe max u v. If one could quantify over finitely many universes, the above proof would work fine, but this is not the case. One could still write the proof considering spaces in any universe in u, v, w, max u v, max v w, max u v w, but it would be extremely tedious and lead to a lot of duplication. Instead, we formulate the above proof when all spaces live in the same universe (where everything is fine), and then we deduce the general result by lifting all our spaces to a common universe. We use the trick that any space H is isomorphic through a continuous linear equiv to continuous_multilinear_map (Ξ» (i : fin 0), E Γ— F Γ— G) H to change the universe level, and then argue that composing with such a linear equiv does not change the fact of being C^n, which we have already proved previously.

theorem times_cont_diff_on.comp {π•œ : Type u_1} [nondiscrete_normed_field π•œ] {E : Type u_2} [normed_group E] [normed_space π•œ E] {F : Type u_3} [normed_group F] [normed_space π•œ F] {G : Type u_4} [normed_group G] [normed_space π•œ G] {n : with_top β„•} {s : set E} {t : set F} {g : F β†’ G} {f : E β†’ F} :
times_cont_diff_on π•œ n g t β†’ times_cont_diff_on π•œ n f s β†’ s βŠ† f ⁻¹' t β†’ times_cont_diff_on π•œ n (g ∘ f) s

The composition of C^n functions on domains is C^n.

theorem times_cont_diff_on.comp' {π•œ : Type u_1} [nondiscrete_normed_field π•œ] {E : Type u_2} [normed_group E] [normed_space π•œ E] {F : Type u_3} [normed_group F] [normed_space π•œ F] {G : Type u_4} [normed_group G] [normed_space π•œ G] {n : with_top β„•} {s : set E} {t : set F} {g : F β†’ G} {f : E β†’ F} :
times_cont_diff_on π•œ n g t β†’ times_cont_diff_on π•œ n f s β†’ times_cont_diff_on π•œ n (g ∘ f) (s ∩ f ⁻¹' t)

The composition of C^n functions on domains is C^n.

theorem times_cont_diff.comp_times_cont_diff_on {π•œ : Type u_1} [nondiscrete_normed_field π•œ] {E : Type u_2} [normed_group E] [normed_space π•œ E] {F : Type u_3} [normed_group F] [normed_space π•œ F] {G : Type u_4} [normed_group G] [normed_space π•œ G] {n : with_top β„•} {s : set E} {g : F β†’ G} {f : E β†’ F} :
times_cont_diff π•œ n g β†’ times_cont_diff_on π•œ n f s β†’ times_cont_diff_on π•œ n (g ∘ f) s

The composition of a C^n function on a domain with a C^n function is C^n.

theorem times_cont_diff.comp {π•œ : Type u_1} [nondiscrete_normed_field π•œ] {E : Type u_2} [normed_group E] [normed_space π•œ E] {F : Type u_3} [normed_group F] [normed_space π•œ F] {G : Type u_4} [normed_group G] [normed_space π•œ G] {n : with_top β„•} {g : F β†’ G} {f : E β†’ F} :
times_cont_diff π•œ n g β†’ times_cont_diff π•œ n f β†’ times_cont_diff π•œ n (g ∘ f)

The composition of C^n functions is C^n.

theorem times_cont_diff_within_at.comp {π•œ : Type u_1} [nondiscrete_normed_field π•œ] {E : Type u_2} [normed_group E] [normed_space π•œ E] {F : Type u_3} [normed_group F] [normed_space π•œ F] {G : Type u_4} [normed_group G] [normed_space π•œ G] {n : with_top β„•} {s : set E} {t : set F} {g : F β†’ G} {f : E β†’ F} (x : E) :
times_cont_diff_within_at π•œ n g t (f x) β†’ times_cont_diff_within_at π•œ n f s x β†’ s βŠ† f ⁻¹' t β†’ times_cont_diff_within_at π•œ n (g ∘ f) s x

The composition of C^n functions at points in domains is C^n.

theorem times_cont_diff_within_at.comp' {π•œ : Type u_1} [nondiscrete_normed_field π•œ] {E : Type u_2} [normed_group E] [normed_space π•œ E] {F : Type u_3} [normed_group F] [normed_space π•œ F] {G : Type u_4} [normed_group G] [normed_space π•œ G] {n : with_top β„•} {s : set E} {t : set F} {g : F β†’ G} {f : E β†’ F} (x : E) :
times_cont_diff_within_at π•œ n g t (f x) β†’ times_cont_diff_within_at π•œ n f s x β†’ times_cont_diff_within_at π•œ n (g ∘ f) (s ∩ f ⁻¹' t) x

The composition of C^n functions at points in domains is C^n.

theorem times_cont_diff_at.comp {π•œ : Type u_1} [nondiscrete_normed_field π•œ] {E : Type u_2} [normed_group E] [normed_space π•œ E] {F : Type u_3} [normed_group F] [normed_space π•œ F] {G : Type u_4} [normed_group G] [normed_space π•œ G] {n : with_top β„•} {g : F β†’ G} {f : E β†’ F} (x : E) :
times_cont_diff_at π•œ n g (f x) β†’ times_cont_diff_at π•œ n f x β†’ times_cont_diff_at π•œ n (g ∘ f) x

The composition of C^n functions at points is C^n.

theorem times_cont_diff.comp_times_cont_diff_within_at {π•œ : Type u_1} [nondiscrete_normed_field π•œ] {E : Type u_2} [normed_group E] [normed_space π•œ E] {F : Type u_3} [normed_group F] [normed_space π•œ F] {G : Type u_4} [normed_group G] [normed_space π•œ G] {t : set E} {x : E} {n : with_top β„•} {g : F β†’ G} {f : E β†’ F} :
times_cont_diff π•œ n g β†’ times_cont_diff_within_at π•œ n f t x β†’ times_cont_diff_within_at π•œ n (g ∘ f) t x

theorem times_cont_diff.comp_times_cont_diff_at {π•œ : Type u_1} [nondiscrete_normed_field π•œ] {E : Type u_2} [normed_group E] [normed_space π•œ E] {F : Type u_3} [normed_group F] [normed_space π•œ F] {G : Type u_4} [normed_group G] [normed_space π•œ G] {n : with_top β„•} {g : F β†’ G} {f : E β†’ F} (x : E) :
times_cont_diff π•œ n g β†’ times_cont_diff_at π•œ n f x β†’ times_cont_diff_at π•œ n (g ∘ f) x

theorem times_cont_diff_on_fderiv_within_apply {π•œ : Type u_1} [nondiscrete_normed_field π•œ] {E : Type u_2} [normed_group E] [normed_space π•œ E] {F : Type u_3} [normed_group F] [normed_space π•œ F] {m n : with_top β„•} {s : set E} {f : E β†’ F} :
times_cont_diff_on π•œ n f s β†’ unique_diff_on π•œ s β†’ m + 1 ≀ n β†’ times_cont_diff_on π•œ m (Ξ» (p : E Γ— E), ⇑(fderiv_within π•œ f s p.fst) p.snd) (s.prod set.univ)

The bundled derivative of a C^{n+1} function is C^n.

theorem times_cont_diff.times_cont_diff_fderiv_apply {π•œ : Type u_1} [nondiscrete_normed_field π•œ] {E : Type u_2} [normed_group E] [normed_space π•œ E] {F : Type u_3} [normed_group F] [normed_space π•œ F] {n m : with_top β„•} {f : E β†’ F} :
times_cont_diff π•œ n f β†’ m + 1 ≀ n β†’ times_cont_diff π•œ m (Ξ» (p : E Γ— E), ⇑(fderiv π•œ f p.fst) p.snd)

The bundled derivative of a C^{n+1} function is C^n.

Sum of two functions

theorem times_cont_diff_add {π•œ : Type u_1} [nondiscrete_normed_field π•œ] {F : Type u_3} [normed_group F] [normed_space π•œ F] {n : with_top β„•} :
times_cont_diff π•œ n (Ξ» (p : F Γ— F), p.fst + p.snd)

theorem times_cont_diff_within_at.add {π•œ : Type u_1} [nondiscrete_normed_field π•œ] {E : Type u_2} [normed_group E] [normed_space π•œ E] {F : Type u_3} [normed_group F] [normed_space π•œ F] {x : E} {n : with_top β„•} {s : set E} {f g : E β†’ F} :
times_cont_diff_within_at π•œ n f s x β†’ times_cont_diff_within_at π•œ n g s x β†’ times_cont_diff_within_at π•œ n (Ξ» (x : E), f x + g x) s x

The sum of two C^n functions within a set at a point is C^n within this set at this point.

theorem times_cont_diff_at.add {π•œ : Type u_1} [nondiscrete_normed_field π•œ] {E : Type u_2} [normed_group E] [normed_space π•œ E] {F : Type u_3} [normed_group F] [normed_space π•œ F] {x : E} {n : with_top β„•} {f g : E β†’ F} :
times_cont_diff_at π•œ n f x β†’ times_cont_diff_at π•œ n g x β†’ times_cont_diff_at π•œ n (Ξ» (x : E), f x + g x) x

The sum of two C^n functions at a point is C^n at this point.

theorem times_cont_diff.add {π•œ : Type u_1} [nondiscrete_normed_field π•œ] {E : Type u_2} [normed_group E] [normed_space π•œ E] {F : Type u_3} [normed_group F] [normed_space π•œ F] {n : with_top β„•} {f g : E β†’ F} :
times_cont_diff π•œ n f β†’ times_cont_diff π•œ n g β†’ times_cont_diff π•œ n (Ξ» (x : E), f x + g x)

The sum of two C^nfunctions is C^n.

theorem times_cont_diff_on.add {π•œ : Type u_1} [nondiscrete_normed_field π•œ] {E : Type u_2} [normed_group E] [normed_space π•œ E] {F : Type u_3} [normed_group F] [normed_space π•œ F] {n : with_top β„•} {s : set E} {f g : E β†’ F} :
times_cont_diff_on π•œ n f s β†’ times_cont_diff_on π•œ n g s β†’ times_cont_diff_on π•œ n (Ξ» (x : E), f x + g x) s

The sum of two C^n functions on a domain is C^n.

Negative

theorem times_cont_diff_neg {π•œ : Type u_1} [nondiscrete_normed_field π•œ] {F : Type u_3} [normed_group F] [normed_space π•œ F] {n : with_top β„•} :
times_cont_diff π•œ n (Ξ» (p : F), -p)

theorem times_cont_diff_within_at.neg {π•œ : Type u_1} [nondiscrete_normed_field π•œ] {E : Type u_2} [normed_group E] [normed_space π•œ E] {F : Type u_3} [normed_group F] [normed_space π•œ F] {x : E} {n : with_top β„•} {s : set E} {f : E β†’ F} :
times_cont_diff_within_at π•œ n f s x β†’ times_cont_diff_within_at π•œ n (Ξ» (x : E), -f x) s x

The negative of a C^n function within a domain at a point is C^n within this domain at this point.

theorem times_cont_diff_at.neg {π•œ : Type u_1} [nondiscrete_normed_field π•œ] {E : Type u_2} [normed_group E] [normed_space π•œ E] {F : Type u_3} [normed_group F] [normed_space π•œ F] {x : E} {n : with_top β„•} {f : E β†’ F} :
times_cont_diff_at π•œ n f x β†’ times_cont_diff_at π•œ n (Ξ» (x : E), -f x) x

The negative of a C^n function at a point is C^n at this point.

theorem times_cont_diff.neg {π•œ : Type u_1} [nondiscrete_normed_field π•œ] {E : Type u_2} [normed_group E] [normed_space π•œ E] {F : Type u_3} [normed_group F] [normed_space π•œ F] {n : with_top β„•} {f : E β†’ F} :
times_cont_diff π•œ n f β†’ times_cont_diff π•œ n (Ξ» (x : E), -f x)

The negative of a C^nfunction is C^n.

theorem times_cont_diff_on.neg {π•œ : Type u_1} [nondiscrete_normed_field π•œ] {E : Type u_2} [normed_group E] [normed_space π•œ E] {F : Type u_3} [normed_group F] [normed_space π•œ F] {n : with_top β„•} {s : set E} {f : E β†’ F} :
times_cont_diff_on π•œ n f s β†’ times_cont_diff_on π•œ n (Ξ» (x : E), -f x) s

The negative of a C^n function on a domain is C^n.

Subtraction

theorem times_cont_diff_within_at.sub {π•œ : Type u_1} [nondiscrete_normed_field π•œ] {E : Type u_2} [normed_group E] [normed_space π•œ E] {F : Type u_3} [normed_group F] [normed_space π•œ F] {x : E} {n : with_top β„•} {s : set E} {f g : E β†’ F} :
times_cont_diff_within_at π•œ n f s x β†’ times_cont_diff_within_at π•œ n g s x β†’ times_cont_diff_within_at π•œ n (Ξ» (x : E), f x - g x) s x

The difference of two C^n functions within a set at a point is C^n within this set at this point.

theorem times_cont_diff_at.sub {π•œ : Type u_1} [nondiscrete_normed_field π•œ] {E : Type u_2} [normed_group E] [normed_space π•œ E] {F : Type u_3} [normed_group F] [normed_space π•œ F] {x : E} {n : with_top β„•} {f g : E β†’ F} :
times_cont_diff_at π•œ n f x β†’ times_cont_diff_at π•œ n g x β†’ times_cont_diff_at π•œ n (Ξ» (x : E), f x - g x) x

The difference of two C^n functions at a point is C^n at this point.

theorem times_cont_diff_on.sub {π•œ : Type u_1} [nondiscrete_normed_field π•œ] {E : Type u_2} [normed_group E] [normed_space π•œ E] {F : Type u_3} [normed_group F] [normed_space π•œ F] {n : with_top β„•} {s : set E} {f g : E β†’ F} :
times_cont_diff_on π•œ n f s β†’ times_cont_diff_on π•œ n g s β†’ times_cont_diff_on π•œ n (Ξ» (x : E), f x - g x) s

The difference of two C^n functions on a domain is C^n.

theorem times_cont_diff.sub {π•œ : Type u_1} [nondiscrete_normed_field π•œ] {E : Type u_2} [normed_group E] [normed_space π•œ E] {F : Type u_3} [normed_group F] [normed_space π•œ F] {n : with_top β„•} {f g : E β†’ F} :
times_cont_diff π•œ n f β†’ times_cont_diff π•œ n g β†’ times_cont_diff π•œ n (Ξ» (x : E), f x - g x)

The difference of two C^n functions is C^n.

Sum of finitely many functions

theorem times_cont_diff_within_at.sum {π•œ : Type u_1} [nondiscrete_normed_field π•œ] {E : Type u_2} [normed_group E] [normed_space π•œ E] {F : Type u_3} [normed_group F] [normed_space π•œ F] {ΞΉ : Type u_4} {f : ΞΉ β†’ E β†’ F} {s : finset ΞΉ} {n : with_top β„•} {t : set E} {x : E} :
(βˆ€ (i : ΞΉ), i ∈ s β†’ times_cont_diff_within_at π•œ n (Ξ» (x : E), f i x) t x) β†’ times_cont_diff_within_at π•œ n (Ξ» (x : E), βˆ‘ (i : ΞΉ) in s, f i x) t x

theorem times_cont_diff_at.sum {π•œ : Type u_1} [nondiscrete_normed_field π•œ] {E : Type u_2} [normed_group E] [normed_space π•œ E] {F : Type u_3} [normed_group F] [normed_space π•œ F] {ΞΉ : Type u_4} {f : ΞΉ β†’ E β†’ F} {s : finset ΞΉ} {n : with_top β„•} {x : E} :
(βˆ€ (i : ΞΉ), i ∈ s β†’ times_cont_diff_at π•œ n (Ξ» (x : E), f i x) x) β†’ times_cont_diff_at π•œ n (Ξ» (x : E), βˆ‘ (i : ΞΉ) in s, f i x) x

theorem times_cont_diff_on.sum {π•œ : Type u_1} [nondiscrete_normed_field π•œ] {E : Type u_2} [normed_group E] [normed_space π•œ E] {F : Type u_3} [normed_group F] [normed_space π•œ F] {ΞΉ : Type u_4} {f : ΞΉ β†’ E β†’ F} {s : finset ΞΉ} {n : with_top β„•} {t : set E} :
(βˆ€ (i : ΞΉ), i ∈ s β†’ times_cont_diff_on π•œ n (Ξ» (x : E), f i x) t) β†’ times_cont_diff_on π•œ n (Ξ» (x : E), βˆ‘ (i : ΞΉ) in s, f i x) t

theorem times_cont_diff.sum {π•œ : Type u_1} [nondiscrete_normed_field π•œ] {E : Type u_2} [normed_group E] [normed_space π•œ E] {F : Type u_3} [normed_group F] [normed_space π•œ F] {ΞΉ : Type u_4} {f : ΞΉ β†’ E β†’ F} {s : finset ΞΉ} {n : with_top β„•} :
(βˆ€ (i : ΞΉ), i ∈ s β†’ times_cont_diff π•œ n (Ξ» (x : E), f i x)) β†’ times_cont_diff π•œ n (Ξ» (x : E), βˆ‘ (i : ΞΉ) in s, f i x)

Product of two functions

theorem times_cont_diff_mul {π•œ : Type u_1} [nondiscrete_normed_field π•œ] {n : with_top β„•} :
times_cont_diff π•œ n (Ξ» (p : π•œ Γ— π•œ), (p.fst) * p.snd)

theorem times_cont_diff_within_at.mul {π•œ : Type u_1} [nondiscrete_normed_field π•œ] {E : Type u_2} [normed_group E] [normed_space π•œ E] {x : E} {n : with_top β„•} {s : set E} {f g : E β†’ π•œ} :
times_cont_diff_within_at π•œ n f s x β†’ times_cont_diff_within_at π•œ n g s x β†’ times_cont_diff_within_at π•œ n (Ξ» (x : E), (f x) * g x) s x

The product of two C^n functions within a set at a point is C^n within this set at this point.

theorem times_cont_diff_at.mul {π•œ : Type u_1} [nondiscrete_normed_field π•œ] {E : Type u_2} [normed_group E] [normed_space π•œ E] {x : E} {n : with_top β„•} {f g : E β†’ π•œ} :
times_cont_diff_at π•œ n f x β†’ times_cont_diff_at π•œ n g x β†’ times_cont_diff_at π•œ n (Ξ» (x : E), (f x) * g x) x

The product of two C^n functions at a point is C^n at this point.

theorem times_cont_diff.mul {π•œ : Type u_1} [nondiscrete_normed_field π•œ] {E : Type u_2} [normed_group E] [normed_space π•œ E] {n : with_top β„•} {f g : E β†’ π•œ} :
times_cont_diff π•œ n f β†’ times_cont_diff π•œ n g β†’ times_cont_diff π•œ n (Ξ» (x : E), (f x) * g x)

The product of two C^nfunctions is C^n.

theorem times_cont_diff_on.mul {π•œ : Type u_1} [nondiscrete_normed_field π•œ] {E : Type u_2} [normed_group E] [normed_space π•œ E] {n : with_top β„•} {s : set E} {f g : E β†’ π•œ} :
times_cont_diff_on π•œ n f s β†’ times_cont_diff_on π•œ n g s β†’ times_cont_diff_on π•œ n (Ξ» (x : E), (f x) * g x) s

The product of two C^n functions on a domain is C^n.

Scalar multiplication

theorem times_cont_diff_smul {π•œ : Type u_1} [nondiscrete_normed_field π•œ] {F : Type u_3} [normed_group F] [normed_space π•œ F] {n : with_top β„•} :
times_cont_diff π•œ n (Ξ» (p : π•œ Γ— F), p.fst β€’ p.snd)

theorem times_cont_diff_within_at.smul {π•œ : Type u_1} [nondiscrete_normed_field π•œ] {E : Type u_2} [normed_group E] [normed_space π•œ E] {F : Type u_3} [normed_group F] [normed_space π•œ F] {x : E} {n : with_top β„•} {s : set E} {f : E β†’ π•œ} {g : E β†’ F} :
times_cont_diff_within_at π•œ n f s x β†’ times_cont_diff_within_at π•œ n g s x β†’ times_cont_diff_within_at π•œ n (Ξ» (x : E), f x β€’ g x) s x

The scalar multiplication of two C^n functions within a set at a point is C^n within this set at this point.

theorem times_cont_diff_at.smul {π•œ : Type u_1} [nondiscrete_normed_field π•œ] {E : Type u_2} [normed_group E] [normed_space π•œ E] {F : Type u_3} [normed_group F] [normed_space π•œ F] {x : E} {n : with_top β„•} {f : E β†’ π•œ} {g : E β†’ F} :
times_cont_diff_at π•œ n f x β†’ times_cont_diff_at π•œ n g x β†’ times_cont_diff_at π•œ n (Ξ» (x : E), f x β€’ g x) x

The scalar multiplication of two C^n functions at a point is C^n at this point.

theorem times_cont_diff.smul {π•œ : Type u_1} [nondiscrete_normed_field π•œ] {E : Type u_2} [normed_group E] [normed_space π•œ E] {F : Type u_3} [normed_group F] [normed_space π•œ F] {n : with_top β„•} {f : E β†’ π•œ} {g : E β†’ F} :
times_cont_diff π•œ n f β†’ times_cont_diff π•œ n g β†’ times_cont_diff π•œ n (Ξ» (x : E), f x β€’ g x)

The scalar multiplication of two C^n functions is C^n.

theorem times_cont_diff_on.smul {π•œ : Type u_1} [nondiscrete_normed_field π•œ] {E : Type u_2} [normed_group E] [normed_space π•œ E] {F : Type u_3} [normed_group F] [normed_space π•œ F] {n : with_top β„•} {s : set E} {f : E β†’ π•œ} {g : E β†’ F} :
times_cont_diff_on π•œ n f s β†’ times_cont_diff_on π•œ n g s β†’ times_cont_diff_on π•œ n (Ξ» (x : E), f x β€’ g x) s

The scalar multiplication of two C^n functions on a domain is C^n.

Cartesian product of two functions

theorem times_cont_diff_within_at.prod_map' {π•œ : Type u_1} [nondiscrete_normed_field π•œ] {E : Type u_2} [normed_group E] [normed_space π•œ E] {F : Type u_3} [normed_group F] [normed_space π•œ F] {E' : Type u_5} [normed_group E'] [normed_space π•œ E'] {F' : Type u_6} [normed_group F'] [normed_space π•œ F'] {n : with_top β„•} {s : set E} {t : set E'} {f : E β†’ F} {g : E' β†’ F'} {p : E Γ— E'} :
times_cont_diff_within_at π•œ n f s p.fst β†’ times_cont_diff_within_at π•œ n g t p.snd β†’ times_cont_diff_within_at π•œ n (prod.map f g) (s.prod t) p

The product map of two C^n functions within a set at a point is C^n within the product set at the product point.

theorem times_cont_diff_within_at.prod_map {π•œ : Type u_1} [nondiscrete_normed_field π•œ] {E : Type u_2} [normed_group E] [normed_space π•œ E] {F : Type u_3} [normed_group F] [normed_space π•œ F] {E' : Type u_5} [normed_group E'] [normed_space π•œ E'] {F' : Type u_6} [normed_group F'] [normed_space π•œ F'] {n : with_top β„•} {s : set E} {t : set E'} {f : E β†’ F} {g : E' β†’ F'} {x : E} {y : E'} :
times_cont_diff_within_at π•œ n f s x β†’ times_cont_diff_within_at π•œ n g t y β†’ times_cont_diff_within_at π•œ n (prod.map f g) (s.prod t) (x, y)

theorem times_cont_diff_on.prod_map {π•œ : Type u_1} [nondiscrete_normed_field π•œ] {E : Type u_2} [normed_group E] [normed_space π•œ E] {F : Type u_3} [normed_group F] [normed_space π•œ F] {E' : Type u_4} [normed_group E'] [normed_space π•œ E'] {F' : Type u_5} [normed_group F'] [normed_space π•œ F'] {s : set E} {t : set E'} {n : with_top β„•} {f : E β†’ F} {g : E' β†’ F'} :
times_cont_diff_on π•œ n f s β†’ times_cont_diff_on π•œ n g t β†’ times_cont_diff_on π•œ n (prod.map f g) (s.prod t)

The product map of two C^n functions on a set is C^n on the product set.

theorem times_cont_diff_at.prod_map {π•œ : Type u_1} [nondiscrete_normed_field π•œ] {E : Type u_2} [normed_group E] [normed_space π•œ E] {F : Type u_3} [normed_group F] [normed_space π•œ F] {E' : Type u_5} [normed_group E'] [normed_space π•œ E'] {F' : Type u_6} [normed_group F'] [normed_space π•œ F'] {n : with_top β„•} {f : E β†’ F} {g : E' β†’ F'} {x : E} {y : E'} :
times_cont_diff_at π•œ n f x β†’ times_cont_diff_at π•œ n g y β†’ times_cont_diff_at π•œ n (prod.map f g) (x, y)

The product map of two C^n functions within a set at a point is C^n within the product set at the product point.

theorem times_cont_diff_at.prod_map' {π•œ : Type u_1} [nondiscrete_normed_field π•œ] {E : Type u_2} [normed_group E] [normed_space π•œ E] {F : Type u_3} [normed_group F] [normed_space π•œ F] {E' : Type u_5} [normed_group E'] [normed_space π•œ E'] {F' : Type u_6} [normed_group F'] [normed_space π•œ F'] {n : with_top β„•} {f : E β†’ F} {g : E' β†’ F'} {p : E Γ— E'} :
times_cont_diff_at π•œ n f p.fst β†’ times_cont_diff_at π•œ n g p.snd β†’ times_cont_diff_at π•œ n (prod.map f g) p

The product map of two C^n functions within a set at a point is C^n within the product set at the product point.

theorem times_cont_diff.prod_map {π•œ : Type u_1} [nondiscrete_normed_field π•œ] {E : Type u_2} [normed_group E] [normed_space π•œ E] {F : Type u_3} [normed_group F] [normed_space π•œ F] {E' : Type u_5} [normed_group E'] [normed_space π•œ E'] {F' : Type u_6} [normed_group F'] [normed_space π•œ F'] {n : with_top β„•} {f : E β†’ F} {g : E' β†’ F'} :
times_cont_diff π•œ n f β†’ times_cont_diff π•œ n g β†’ times_cont_diff π•œ n (prod.map f g)

The product map of two C^n functions is C^n.

Inversion in a complete normed algebra

theorem times_cont_diff_at_ring_inverse (π•œ : Type u_1) [nondiscrete_normed_field π•œ] (R : Type u_5) [normed_ring R] [normed_algebra π•œ R] [complete_space R] {n : with_top β„•} (x : units R) :

In a complete normed algebra, the operation of inversion is C^n, for all n, at each invertible element. The proof is by induction, bootstrapping using an identity expressing the derivative of inversion as a bilinear map of inversion itself.

Inversion of continuous linear maps between Banach spaces

theorem times_cont_diff_at_map_inverse {π•œ : Type u_1} [nondiscrete_normed_field π•œ] {E : Type u_2} [normed_group E] [normed_space π•œ E] {F : Type u_3} [normed_group F] [normed_space π•œ F] [complete_space E] {n : with_top β„•} (e : E ≃L[π•œ] F) :

At a continuous linear equivalence e : E ≃L[π•œ] F between Banach spaces, the operation of inversion is C^n, for all n.

theorem times_cont_diff_at.of_local_homeomorph {π•œ : Type u_1} [nondiscrete_normed_field π•œ] {E : Type u_2} [normed_group E] [normed_space π•œ E] {F : Type u_3} [normed_group F] [normed_space π•œ F] [complete_space E] {n : with_top β„•} {f : local_homeomorph E F} {fβ‚€' : E ≃L[π•œ] F} {a : F} :
a ∈ f.to_local_equiv.target β†’ has_fderiv_at ⇑f ↑fβ‚€' (⇑(f.symm) a) β†’ times_cont_diff_at π•œ n ⇑f (⇑(f.symm) a) β†’ times_cont_diff_at π•œ n ⇑(f.symm) a

If f is a local homeomorphism and the point a is in its target, and if f is n times continuously differentiable at f.symm a, and if the derivative at f.symm a is a continuous linear equivalence, then f.symm is n times continuously differentiable at the point a.

This is one of the easy parts of the inverse function theorem: it assumes that we already have an inverse function.

Results over ℝ

The results in this section rely on the Mean Value Theorem, and therefore hold only over ℝ (and its extension fields such as β„‚).

theorem has_ftaylor_series_up_to_on.has_strict_fderiv_at {E' : Type u_5} [normed_group E'] [normed_space ℝ E'] {F' : Type u_6} [normed_group F'] [normed_space ℝ F'] {s : set E'} {f : E' β†’ F'} {x : E'} {p : E' β†’ formal_multilinear_series ℝ E' F'} {n : with_top β„•} :

If a function has a Taylor series at order at least 1, then at points in the interior of the domain of definition, the term of order 1 of this series is a strict derivative of f.

theorem times_cont_diff_at.has_strict_fderiv_at {E' : Type u_5} [normed_group E'] [normed_space ℝ E'] {F' : Type u_6} [normed_group F'] [normed_space ℝ F'] {f : E' β†’ F'} {x : E'} {n : with_top β„•} :

If a function is C^n with 1 ≀ n around a point, then the derivative of f at this point is also a strict derivative.

theorem times_cont_diff_at.has_strict_fderiv_at' {E' : Type u_5} [normed_group E'] [normed_space ℝ E'] {F' : Type u_6} [normed_group F'] [normed_space ℝ F'] {f : E' β†’ F'} {f' : E' β†’L[ℝ] F'} {x : E'} {n : with_top β„•} :
times_cont_diff_at ℝ n f x β†’ has_fderiv_at f f' x β†’ 1 ≀ n β†’ has_strict_fderiv_at f f' x

If a function is C^n with 1 ≀ n around a point, and its derivative at that point is given to us as f', then f' is also a strict derivative.

theorem times_cont_diff.has_strict_fderiv_at {E' : Type u_5} [normed_group E'] [normed_space ℝ E'] {F' : Type u_6} [normed_group F'] [normed_space ℝ F'] {f : E' β†’ F'} {x : E'} {n : with_top β„•} :

If a function is C^n with 1 ≀ n, then the derivative of f is also a strict derivative.

One dimension

All results up to now have been expressed in terms of the general FrΓ©chet derivative fderiv. For maps defined on the field, the one-dimensional derivative deriv is often easier to use. In this paragraph, we reformulate some higher smoothness results in terms of deriv.

theorem times_cont_diff_on_succ_iff_deriv_within {π•œ : Type u_1} [nondiscrete_normed_field π•œ] {F : Type u_3} [normed_group F] [normed_space π•œ F] {fβ‚‚ : π•œ β†’ F} {sβ‚‚ : set π•œ} {n : β„•} :
unique_diff_on π•œ sβ‚‚ β†’ (times_cont_diff_on π•œ ↑(n + 1) fβ‚‚ sβ‚‚ ↔ differentiable_on π•œ fβ‚‚ sβ‚‚ ∧ times_cont_diff_on π•œ ↑n (deriv_within fβ‚‚ sβ‚‚) sβ‚‚)

A function is C^(n + 1) on a domain with unique derivatives if and only if it is differentiable there, and its derivative (formulated with deriv_within) is C^n.

theorem times_cont_diff_on_succ_iff_deriv_of_open {π•œ : Type u_1} [nondiscrete_normed_field π•œ] {F : Type u_3} [normed_group F] [normed_space π•œ F] {fβ‚‚ : π•œ β†’ F} {sβ‚‚ : set π•œ} {n : β„•} :
is_open sβ‚‚ β†’ (times_cont_diff_on π•œ ↑(n + 1) fβ‚‚ sβ‚‚ ↔ differentiable_on π•œ fβ‚‚ sβ‚‚ ∧ times_cont_diff_on π•œ ↑n (deriv fβ‚‚) sβ‚‚)

A function is C^(n + 1) on an open domain if and only if it is differentiable there, and its derivative (formulated with deriv) is C^n.

theorem times_cont_diff_on_top_iff_deriv_within {π•œ : Type u_1} [nondiscrete_normed_field π•œ] {F : Type u_3} [normed_group F] [normed_space π•œ F] {fβ‚‚ : π•œ β†’ F} {sβ‚‚ : set π•œ} :
unique_diff_on π•œ sβ‚‚ β†’ (times_cont_diff_on π•œ ⊀ fβ‚‚ sβ‚‚ ↔ differentiable_on π•œ fβ‚‚ sβ‚‚ ∧ times_cont_diff_on π•œ ⊀ (deriv_within fβ‚‚ sβ‚‚) sβ‚‚)

A function is C^∞ on a domain with unique derivatives if and only if it is differentiable there, and its derivative (formulated with deriv_within) is C^∞.

theorem times_cont_diff_on_top_iff_deriv_of_open {π•œ : Type u_1} [nondiscrete_normed_field π•œ] {F : Type u_3} [normed_group F] [normed_space π•œ F] {fβ‚‚ : π•œ β†’ F} {sβ‚‚ : set π•œ} :
is_open sβ‚‚ β†’ (times_cont_diff_on π•œ ⊀ fβ‚‚ sβ‚‚ ↔ differentiable_on π•œ fβ‚‚ sβ‚‚ ∧ times_cont_diff_on π•œ ⊀ (deriv fβ‚‚) sβ‚‚)

A function is C^∞ on an open domain if and only if it is differentiable there, and its derivative (formulated with deriv) is C^∞.

theorem times_cont_diff_on.deriv_within {π•œ : Type u_1} [nondiscrete_normed_field π•œ] {F : Type u_3} [normed_group F] [normed_space π•œ F] {fβ‚‚ : π•œ β†’ F} {sβ‚‚ : set π•œ} {m n : with_top β„•} :
times_cont_diff_on π•œ n fβ‚‚ sβ‚‚ β†’ unique_diff_on π•œ sβ‚‚ β†’ m + 1 ≀ n β†’ times_cont_diff_on π•œ m (deriv_within fβ‚‚ sβ‚‚) sβ‚‚

theorem times_cont_diff_on.deriv_of_open {π•œ : Type u_1} [nondiscrete_normed_field π•œ] {F : Type u_3} [normed_group F] [normed_space π•œ F] {fβ‚‚ : π•œ β†’ F} {sβ‚‚ : set π•œ} {m n : with_top β„•} :
times_cont_diff_on π•œ n fβ‚‚ sβ‚‚ β†’ is_open sβ‚‚ β†’ m + 1 ≀ n β†’ times_cont_diff_on π•œ m (deriv fβ‚‚) sβ‚‚

theorem times_cont_diff_on.continuous_on_deriv_within {π•œ : Type u_1} [nondiscrete_normed_field π•œ] {F : Type u_3} [normed_group F] [normed_space π•œ F] {fβ‚‚ : π•œ β†’ F} {sβ‚‚ : set π•œ} {n : with_top β„•} :
times_cont_diff_on π•œ n fβ‚‚ sβ‚‚ β†’ unique_diff_on π•œ sβ‚‚ β†’ 1 ≀ n β†’ continuous_on (deriv_within fβ‚‚ sβ‚‚) sβ‚‚

theorem times_cont_diff_on.continuous_on_deriv_of_open {π•œ : Type u_1} [nondiscrete_normed_field π•œ] {F : Type u_3} [normed_group F] [normed_space π•œ F] {fβ‚‚ : π•œ β†’ F} {sβ‚‚ : set π•œ} {n : with_top β„•} :
times_cont_diff_on π•œ n fβ‚‚ sβ‚‚ β†’ is_open sβ‚‚ β†’ 1 ≀ n β†’ continuous_on (deriv fβ‚‚) sβ‚‚