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Suppose $f(x)$ is continuous on the closed interval $[a,b]$. Define $m(x)=\max_{a\leq s\leq x}\, f(s)$, $a\leq x\leq b$. Is $m(x)$ continuous necessarily? Thank you.

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    Yes it is. What are your thoughts?2012-10-14
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    Thank you. I know that $m(x)$ is increasing. So it is continuous unless it has jumps around some points. And this is not very possible since $f(x)$ is continuous. But I just do not know how to show this rigorously. Do you have any suggestion?2012-10-14

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