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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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    The map from R to R factor through circle. As continuous functions on compact set have the two properties, so do periodic functions.2012-12-12

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