Let $f:[a,b]\to \mathbb{R}^n$ be an absolutely continuous function. Is it uniformly continuous?
I know that if $n=1$ it is true and it's called Heine's theorem, but what about $n\geq 1$?
2026-03-31 12:11:11.1774959071
Absolute continuous implies uniform continuous?
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Let $f:[a,b]\longrightarrow\mathbb{R}^{n}$ be absolutely continuous. Then $f$ is continuous. Since $[a,b]$ is compact by the Heine–Borel theorem, $f$ is uniformly continuous.