# mathlib3documentation

analysis.calculus.dslope

# Slope of a differentiable function #

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Given a function f : π β E from a nontrivially normed field to a normed space over this field, dslope f a b is defined as slope f a b = (b - a)β»ΒΉ β’ (f b - f a) for a β  b and as deriv f a for a = b.

In this file we define dslope and prove some basic lemmas about its continuity and differentiability.

noncomputable def dslope {π : Type u_1} {E : Type u_2} [normed_space π E] (f : π β E) (a : π) :
π β E

dslope f a b is defined as slope f a b = (b - a)β»ΒΉ β’ (f b - f a) for a β  b and deriv f a for a = b.

Equations
@[simp]
theorem dslope_same {π : Type u_1} {E : Type u_2} [normed_space π E] (f : π β E) (a : π) :
a a = a
theorem dslope_of_ne {π : Type u_1} {E : Type u_2} [normed_space π E] {a b : π} (f : π β E) (h : b β  a) :
a b = a b
theorem continuous_linear_map.dslope_comp {π : Type u_1} {E : Type u_2} [normed_space π E] {F : Type u_3} [normed_space π F] (f : E βL[π] F) (g : π β E) (a b : π) (H : a = b β g a) :
dslope (βf β g) a b = βf (dslope g a b)
theorem eq_on_dslope_slope {π : Type u_1} {E : Type u_2} [normed_space π E] (f : π β E) (a : π) :
set.eq_on (dslope f a) (slope f a) {a}αΆ
theorem dslope_eventually_eq_slope_of_ne {π : Type u_1} {E : Type u_2} [normed_space π E] {a b : π} (f : π β E) (h : b β  a) :
theorem dslope_eventually_eq_slope_punctured_nhds {π : Type u_1} {E : Type u_2} [normed_space π E] {a : π} (f : π β E) :
@[simp]
theorem sub_smul_dslope {π : Type u_1} {E : Type u_2} [normed_space π E] (f : π β E) (a b : π) :
(b - a) β’ a b = f b - f a
theorem dslope_sub_smul_of_ne {π : Type u_1} {E : Type u_2} [normed_space π E] {a b : π} (f : π β E) (h : b β  a) :
dslope (Ξ» (x : π), (x - a) β’ f x) a b = f b
theorem eq_on_dslope_sub_smul {π : Type u_1} {E : Type u_2} [normed_space π E] (f : π β E) (a : π) :
set.eq_on (dslope (Ξ» (x : π), (x - a) β’ f x) a) f {a}αΆ
theorem dslope_sub_smul {π : Type u_1} {E : Type u_2} [normed_space π E] [decidable_eq π] (f : π β E) (a : π) :
dslope (Ξ» (x : π), (x - a) β’ f x) a = (deriv (Ξ» (x : π), (x - a) β’ f x) a)
@[simp]
theorem continuous_at_dslope_same {π : Type u_1} {E : Type u_2} [normed_space π E] {f : π β E} {a : π} :
theorem continuous_within_at.of_dslope {π : Type u_1} {E : Type u_2} [normed_space π E] {f : π β E} {a b : π} {s : set π} (h : s b) :
b
theorem continuous_at.of_dslope {π : Type u_1} {E : Type u_2} [normed_space π E] {f : π β E} {a b : π} (h : continuous_at (dslope f a) b) :
theorem continuous_on.of_dslope {π : Type u_1} {E : Type u_2} [normed_space π E] {f : π β E} {a : π} {s : set π} (h : continuous_on (dslope f a) s) :
theorem continuous_within_at_dslope_of_ne {π : Type u_1} {E : Type u_2} [normed_space π E] {f : π β E} {a b : π} {s : set π} (h : b β  a) :
s b β b
theorem continuous_at_dslope_of_ne {π : Type u_1} {E : Type u_2} [normed_space π E] {f : π β E} {a b : π} (h : b β  a) :
theorem continuous_on_dslope {π : Type u_1} {E : Type u_2} [normed_space π E] {f : π β E} {a : π} {s : set π} (h : s β nhds a) :
theorem differentiable_within_at.of_dslope {π : Type u_1} {E : Type u_2} [normed_space π E] {f : π β E} {a b : π} {s : set π} (h : (dslope f a) s b) :
f s b
theorem differentiable_at.of_dslope {π : Type u_1} {E : Type u_2} [normed_space π E] {f : π β E} {a b : π} (h : (dslope f a) b) :
f b
theorem differentiable_on.of_dslope {π : Type u_1} {E : Type u_2} [normed_space π E] {f : π β E} {a : π} {s : set π} (h : (dslope f a) s) :
f s
theorem differentiable_within_at_dslope_of_ne {π : Type u_1} {E : Type u_2} [normed_space π E] {f : π β E} {a b : π} {s : set π} (h : b β  a) :
(dslope f a) s b β f s b
theorem differentiable_on_dslope_of_nmem {π : Type u_1} {E : Type u_2} [normed_space π E] {f : π β E} {a : π} {s : set π} (h : a β s) :
(dslope f a) s β f s
theorem differentiable_at_dslope_of_ne {π : Type u_1} {E : Type u_2} [normed_space π E] {f : π β E} {a b : π} (h : b β  a) :
(dslope f a) b β f b