The Fréchet derivative #

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Let E and F be normed spaces, f : E → F, and f' : E →L[𝕜] F a continuous 𝕜-linear map, where 𝕜 is a non-discrete normed field. Then

has_fderiv_within_at f f' s x

says that f has derivative f' at x, where the domain of interest is restricted to s. We also have

has_fderiv_at f f' x := has_fderiv_within_at f f' x univ

Finally,

has_strict_fderiv_at f f' x

means that f : E → F has derivative f' : E →L[𝕜] F in the sense of strict differentiability, i.e., f y - f z - f'(y - z) = o(y - z) as y, z → x. This notion is used in the inverse function theorem, and is defined here only to avoid proving theorems like is_bounded_bilinear_map.has_fderiv_at twice: first for has_fderiv_at, then for has_strict_fderiv_at.

Main results #

In addition to the definition and basic properties of the derivative, the folder analysis/calculus/fderiv/ contains the usual formulas (and existence assertions) for the derivative of

• constants
• the identity
• bounded linear maps (linear.lean)
• bounded bilinear maps (bilinear.lean)
• sum of two functions (add.lean)
• sum of finitely many functions (add.lean)
• multiplication of a function by a scalar constant (add.lean)
• negative of a function (add.lean)
• subtraction of two functions (add.lean)
• multiplication of a function by a scalar function (mul.lean)
• multiplication of two scalar functions (mul.lean)
• composition of functions (the chain rule) (comp.lean)
• inverse function (mul.lean) (assuming that it exists; the inverse function theorem is in ../inverse.lean)

For most binary operations we also define const_op and op_const theorems for the cases when the first or second argument is a constant. This makes writing chains of has_deriv_at's easier, and they more frequently lead to the desired result.

One can also interpret the derivative of a function f : 𝕜 → E as an element of E (by identifying a linear function from 𝕜 to E with its value at 1). Results on the Fréchet derivative are translated to this more elementary point of view on the derivative in the file deriv.lean. The derivative of polynomials is handled there, as it is naturally one-dimensional.

The simplifier is set up to prove automatically that some functions are differentiable, or differentiable at a point (but not differentiable on a set or within a set at a point, as checking automatically that the good domains are mapped one to the other when using composition is not something the simplifier can easily do). This means that one can write example (x : ℝ) : differentiable ℝ (λ x, sin (exp (3 + x^2)) - 5 * cos x) := by simp. If there are divisions, one needs to supply to the simplifier proofs that the denominators do not vanish, as in

example (x : ℝ) (h : 1 + sin x ≠ 0) : differentiable_at ℝ (λ x, exp x / (1 + sin x)) x :=
by simp [h]

Of course, these examples only work once exp, cos and sin have been shown to be differentiable, in analysis.special_functions.trigonometric.

The simplifier is not set up to compute the Fréchet derivative of maps (as these are in general complicated multidimensional linear maps), but it will compute one-dimensional derivatives, see deriv.lean.

Implementation details #

The derivative is defined in terms of the is_o relation, but also characterized in terms of the tendsto relation.

We also introduce predicates differentiable_within_at 𝕜 f s x (where 𝕜 is the base field, f the function to be differentiated, x the point at which the derivative is asserted to exist, and s the set along which the derivative is defined), as well as differentiable_at 𝕜 f x, differentiable_on 𝕜 f s and differentiable 𝕜 f to express the existence of a derivative.

To be able to compute with derivatives, we write fderiv_within 𝕜 f s x and fderiv 𝕜 f x for some choice of a derivative if it exists, and the zero function otherwise. This choice only behaves well along sets for which the derivative is unique, i.e., those for which the tangent directions span a dense subset of the whole space. The predicates unique_diff_within_at s x and unique_diff_on s, defined in tangent_cone.lean express this property. We prove that indeed they imply the uniqueness of the derivative. This is satisfied for open subsets, and in particular for univ. This uniqueness only holds when the field is non-discrete, which we request at the very beginning: otherwise, a derivative can be defined, but it has no interesting properties whatsoever.

To make sure that the simplifier can prove automatically that functions are differentiable, we tag many lemmas with the simp attribute, for instance those saying that the sum of differentiable functions is differentiable, as well as their product, their cartesian product, and so on. A notable exception is the chain rule: we do not mark as a simp lemma the fact that, if f and g are differentiable, then their composition also is: simp would always be able to match this lemma, by taking f or g to be the identity. Instead, for every reasonable function (say, exp), we add a lemma that if f is differentiable then so is (λ x, exp (f x)). This means adding some boilerplate lemmas, but these can also be useful in their own right.

Tests for this ability of the simplifier (with more examples) are provided in tests/differentiable.lean.

Tags #

derivative, differentiable, Fréchet, calculus

Basic properties of the derivative #

Deducing continuity from differentiability #

congr properties of the derivative #