differentiableWithinAt_congr_set — Mathlib

English

If differentiableWithinAt h at x and s ≈ t near x, then DifferentiableWithinAt f on s equals on t under suitable conditions.

Русский

Если differentiableWithinAt имеет место в x и s ближе t, то дифференцируемость сохраняется между s и t.

LaTeX

$$$DifferentiableWithinAt 𝕜 c s x \\Rightarrow DifferentiableWithinAt 𝕜 c t x \\text{ when } s=ᶠ_{𝓝 x} t.$$$

Lean4

theorem differentiableWithinAt_congr_set (h : s =ᶠ[𝓝 x] t) :
    DifferentiableWithinAt 𝕜 f s x ↔ DifferentiableWithinAt 𝕜 f t x :=
  exists_congr fun _ => hasFDerivWithinAt_congr_set h

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