congr_of_eventuallyEq — Mathlib

English

If g is continuous at f(x) and f is continuous within s at x, and f maps s into the domain of g, then g∘f is continuous within s at x.

Русский

Если g непрерывна в точке f(x) и f непрерывна на s в x, и отображение f отправляет s в область определения g, то g∘f непрерывна на s в x.

LaTeX

$$$ (ContinuousAt\ g\ (f(x))) \land (ContinuousWithinAt\ f\ s\ x) \land (MapsTo\ f\ s\ t) \Rightarrow (ContinuousWithinAt\ (g\circ f)\ s\ x).$$$

Lean4

theorem congr_of_eventuallyEq (h : ContinuousAt f x) (hg : g =ᶠ[𝓝 x] f) : ContinuousAt g x :=
  congr h (EventuallyEq.symm hg)

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