congr_of_eventuallyEq_of_mem — Mathlib

English

If two functions agree near x inside s, then ContDiffWithinAt 𝕜 ω is preserved.

Русский

Если две функции согласованы в окрестности x внутри s, то ContDiffWithinAt 𝕜 ω сохраняется.

LaTeX

$$$ ContDiffWithinAt 𝕜 ω f s x \\iff ContDiffWithinAt 𝕜 ω g s x$$

Lean4

theorem congr_of_eventuallyEq_of_mem (h : ContDiffWithinAt 𝕜 n f s x) (h₁ : f₁ =ᶠ[𝓝[s] x] f) (hx : x ∈ s) :
    ContDiffWithinAt 𝕜 n f₁ s x :=
  h.congr_of_eventuallyEq h₁ <| h₁.self_of_nhdsWithin hx

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