congr_contDiffWithinAt_of_insert — Mathlib

English

If nhdsWithin x s and equality on that neighborhood hold, then ContDiffWithinAt is preserved under equality on s.

Русский

Если есть равенство на окрестности nhdsWithin x s, то ContDiffWithinAt сохраняется при равенстве на s.

LaTeX

$$$ (nhdsWithin x s).EventuallyEq f1 f \\Rightarrow Iff (ContDiffWithinAt 𝕜 n f1 s x) (ContDiffWithinAt 𝕜 n f s x)$$

Lean4

theorem congr_contDiffWithinAt_of_insert (h₁ : f₁ =ᶠ[𝓝[insert x s] x] f) :
    ContDiffWithinAt 𝕜 n f₁ s x ↔ ContDiffWithinAt 𝕜 n f s x :=
  ⟨fun H ↦ H.congr_of_eventuallyEq_insert h₁.symm, fun H ↦ H.congr_of_eventuallyEq_insert h₁⟩

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