# Documentation

Mathlib.Analysis.Calculus.FDeriv.Extend

# Extending differentiability to the boundary #

We investigate how differentiable functions inside a set extend to differentiable functions on the boundary. For this, it suffices that the function and its derivative admit limits there. A general version of this statement is given in has_fderiv_at_boundary_of_tendsto_fderiv.

One-dimensional versions, in which one wants to obtain differentiability at the left endpoint or the right endpoint of an interval, are given in has_deriv_at_interval_left_endpoint_of_tendsto_deriv and has_deriv_at_interval_right_endpoint_of_tendsto_deriv. These versions are formulated in terms of the one-dimensional derivative deriv ℝ f.

theorem has_fderiv_at_boundary_of_tendsto_fderiv {E : Type u_1} [] {F : Type u_2} [] {f : EF} {s : Set E} {x : E} {f' : E →L[] F} (f_diff : ) (s_conv : ) (s_open : ) (f_cont : ∀ (y : E), y ) (h : Filter.Tendsto (fun y => fderiv f y) () (nhds f')) :

If a function f is differentiable in a convex open set and continuous on its closure, and its derivative converges to a limit f' at a point on the boundary, then f is differentiable there with derivative f'.

theorem has_deriv_at_interval_left_endpoint_of_tendsto_deriv {E : Type u_1} [] {s : } {e : E} {a : } {f : E} (f_diff : ) (f_lim : ) (hs : s nhdsWithin a ()) (f_lim' : Filter.Tendsto (fun x => deriv f x) (nhdsWithin a ()) (nhds e)) :

If a function is differentiable on the right of a point a : ℝ, continuous at a, and its derivative also converges at a, then f is differentiable on the right at a.

theorem has_deriv_at_interval_right_endpoint_of_tendsto_deriv {E : Type u_1} [] {s : } {e : E} {a : } {f : E} (f_diff : ) (f_lim : ) (hs : s nhdsWithin a ()) (f_lim' : Filter.Tendsto (fun x => deriv f x) (nhdsWithin a ()) (nhds e)) :

If a function is differentiable on the left of a point a : ℝ, continuous at a, and its derivative also converges at a, then f is differentiable on the left at a.

theorem hasDerivAt_of_hasDerivAt_of_ne {E : Type u_1} [] {f : E} {g : E} {x : } (f_diff : ∀ (y : ), y xHasDerivAt f (g y) y) (hf : ) (hg : ) :
HasDerivAt f (g x) x

If a real function f has a derivative g everywhere but at a point, and f and g are continuous at this point, then g is also the derivative of f at this point.

theorem hasDerivAt_of_hasDerivAt_of_ne' {E : Type u_1} [] {f : E} {g : E} {x : } (f_diff : ∀ (y : ), y xHasDerivAt f (g y) y) (hf : ) (hg : ) (y : ) :
HasDerivAt f (g y) y

If a real function f has a derivative g everywhere but at a point, and f and g are continuous at this point, then g is the derivative of f everywhere.