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Given continuous and Periodic function, How can I prove that it is Uniformly continuous?

Thank you!

2 Answers 2

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Observe that it is uniformly continuous when restricted to a closed interval whose length is a period of your function (or twice that, to simplify things a little bit), and then use periodicity to extend to the whole of $\mathbb R$.

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    Another hint: There is a mighty theorem saying that a function which is continuous on some compact set $X$ is automatically uniformly continuous on $X$.2011-03-03
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a proof by contradiction but you should do it yourself!