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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


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

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