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So using the definition of periodic function which is there exist $p \neq 0$ such that $f(x+p) = f(x)$ for all $x\in \mathbb{R}$. I know that I only need to prove on the interval $[a,b]$ which other part of the function just repeats what it looks like on the interval $[a,b]$. Where do I go from here?

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Are you familiar with the Extreme Value Theorem? That should help you show that it is bounded. –  user35959 Dec 12 '12 at 3:42
Any continuous function is uniformly continuous on a closed bounded interval. –  Tom Oldfield Dec 12 '12 at 3:49
The image of a compact set through a continuous function is compact :) –  madprob Dec 12 '12 at 4:52
The map from R to R factor through circle. As continuous functions on compact set have the two properties, so do periodic functions. –  lee Dec 12 '12 at 5:02

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