congr_of_eventuallyEq — Mathlib

English

If h has MDifferentiableWithinAt f s x and an eventual equality with f, then MDifferentiableWithinAt f1 s x holds.

Русский

Если у h есть MD differentiableWithinAt f s x и f1 эквивалентна f почти повсюду в окрестности x, то MDifferentiableWithinAt f1 s x выполняется.

LaTeX

$$$MDifferentiableWithinAt\ I I'\ f\ s\ x\ f' \land (f_1 =^\! [𝓝[s] x] f) \Rightarrow MDifferentiableWithinAt\ I I'\ f_1\ s\ x$$$

Lean4

theorem congr_of_eventuallyEq (h : MDifferentiableWithinAt I I' f s x) (h₁ : f₁ =ᶠ[𝓝[s] x] f) (hx : f₁ x = f x) :
    MDifferentiableWithinAt I I' f₁ s x :=
  (h.hasMFDerivWithinAt.congr_of_eventuallyEq h₁ hx).mdifferentiableWithinAt

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