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If $f(x)$ is uniformly continuous at $(0,1)$ then is it bounded at $(0,1)$?
Uniform continuity and boundedness

This was a homework assignment I was asked to do:

Let $f: (0,1) \to \mathbb{R} $ be uniformly continuous. Show that $f$ is bounded. (I.e. you have to show that there exist some $m, M \in \mathbb{R}$ such that $ m \leq f(x) \leq M$ for all $x \in (0,1) $ ). I tried proving this using the definition of uniform continuity, but to no avail.

Thanks in advance

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marked as duplicate by t.b., Dylan Moreland, Rudy the Reindeer, Martin Sleziak, Did Dec 15 '11 at 20:13

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