hasDerivWithinAt_congr_set — Mathlib

English

If nhds x and the two sets s,t have equal nhds at x, HasDerivWithinAt f f' s x is equivalent to HasDerivWithinAt f f' t x.

Русский

Если окружности x совпадают по соседству для s и t, то производные по s и t равны.

LaTeX

$$$\\text{hasDerivWithinAt_congr_set }\\; h:\\; (nhds\\,x).EventuallyEq\\; s\\; t \\Rightarrow \\mathrm{HasDerivWithinAt}(f,f',s,x) \\iff \\mathrm{HasDerivWithinAt}(f,f',t,x).$$$

Lean4

theorem hasDerivWithinAt_congr_set {s t : Set 𝕜} (h : s =ᶠ[𝓝 x] t) :
    HasDerivWithinAt f f' s x ↔ HasDerivWithinAt f f' t x :=
  hasFDerivWithinAt_congr_set h

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