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Is this lemma true or false?

Given a function $f : \mathbb R\to\mathbb R$ that is continuous over an open interval $I = ]a,b[$. For each $x\in I$ there is an $\varepsilon>0$ such that $f$ is uniformly continuous in the intervall $[x-\varepsilon,x+\varepsilon] \subset I.$

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    @Arturo I've always wondered what theorem Zorn's Lemma was originally used in...2012-01-17

1 Answers 1

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It is true.

For $x\in(a,b)$, there is a closed, hence compact, set $[x-\epsilon, x+\epsilon]\subset(a,b)$. $f$ is continuous on $[x-\epsilon, x+\epsilon]$ and continuous functions on compact sets are uniformly continuous (see the link here for a proof of this fact).