arsinh — Mathlib

English

If f is continuous within s at a, then arsinh ∘ f is continuous within s at a.

Русский

Если f непрерывна внутри s в точке a, то arsinh(f) непрерывна внутри s в a.

LaTeX

$$$\\forall X\\ {\\text{topological space}},\\ f:\\,X \\to \\mathbb{R},\\ s:\\,\\text{Set }X,\\ a:\\,X,\\ (\\operatorname{ContinuousWithinAt} f s a)\\Rightarrow \\operatorname{ContinuousWithinAt}(\\lambda x. \\operatorname{arsinh}(f x)) s a.$$$

Lean4

nonrec theorem arsinh (h : ContinuousWithinAt f s a) : ContinuousWithinAt (fun x => arsinh (f x)) s a :=
  h.arsinh

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