$$a_n=max_{\{\forall x, x':|f (x)-f (x')|\leq\frac {1}{n}\}}|x-x'|$$ So $a_n $ is the maximal distance between points $x$ and $x'$ such that $|f (x)-f (x')|\leq\frac {1}{n}$ for a strictly increasing continuous $f$. $a_n$ is decreasing and has (as i suppose) limit $0$ as $n\to\infty$, so it seems that there should be some function $w$ such that $$|x-x'|\leq w(|f (x)-f (x')|)$$ What is this $w$?
2026-04-12 03:32:00.1775964720
Relation between difference of values of an increasing function and difference of its arguments
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