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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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By "steady", do you mean "constant", or do you mean "derivative is constant", or this a term of art I am unfamiliar with? You may want to add a definition, and avoid creating a new tag... Especially something as uninformative as [lemma] (Would Zorn's Lemma be tagged as a [lemma], or as a [theorem]?) –  Arturo Magidin Jan 17 '12 at 22:12
    
@Arturo Sorry. That was a translation mistake. In German, one says "stetig" to say "continuous". stetig and steady are false friends... –  FUZxxl Jan 17 '12 at 22:42
    
@Arturo I've always wondered what theorem Zorn's Lemma was originally used in... –  David Mitra Jan 17 '12 at 22:43
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1 Answer

up vote 3 down vote accepted

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).

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