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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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    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

2 Answers 2

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Yes it is.

HINT: Clearly $m(s)$ is monotonically non-decreasing on $[a,b]$. Thus, if it is discontinuous at some $x\in[a,b]$, it must have a jump discontinuity at $x$: either $\lim\limits_{s\to x^-}m(s), or $m(x)<\lim\limits_{s\to x^+}m(s)$.

  1. Suppose first that $\lim\limits_{s\to x^-}m(s) for some $x\in(a,b]$. Show that $\lim\limits_{s\to x^-}f(s), contradicting the continuity of $f$.

  2. Then suppose that $m(x)<\lim\limits_{s\to x^+}m(s)$ for some $x\in[a,b)$. Show that $f(x)<\lim\limits_{s\to x^+}f(s)$, again contradicting the continuity of $f$.

Added: Here’s a little more help with (1). Let $u=\lim\limits_{s\to x^-}m(s)$. For all $s\in[a,x)$ we have $f(s)\le m(s)\le u In other words, $f(s)\le u$ for every $s\in[a,x)$, but $\max\{f(s):a\le s\le x\}=m(x)>u$; this is possible only if $f(x)=m(x)$. But then $\lim\limits_{s\to x^-}f(s)\le u.

(2) can be dealt with very similarly.

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    @Knightgu: You’re welcome; glad it helps.2012-10-14
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$f$ is uniformly continuous on $[a,b]$. Fix $\varepsilon>0$, choose $\delta$ such that $|f(x)-f(y)|\leq\varepsilon$ when $|x-y|\leq \delta$ and let $x\in [a,b]$. We have to show that $|m(x)-m(x+t)|\leq\varepsilon$ if $|t|\leq \delta$. We have for $t>0$ that $m(x+t)=\max_{0\leq s\leq x+t}f(s)=\max\{f(x),\max_{x\leq s\leq x+t}f(s)\},$ and for $t<0$ that $m(x+t)=\max\{f(x-t),\max_{x-t\leq s\leq x}f(s)\}.$ We have $|f(x)-\max_{x\leq s\leq x+t}f(s)\}|\leq\varepsilon$ when $0 and $|f(x-t)-\max_{x-t\leq s\leq x}f(s)|<\varepsilon$ when $0>t>-\delta$.

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    I wonder if it is the case in your answer that $m(x+t)=max\{m(x),\,max_{x\leq s\leq x+t}\,f(s)\}$for t>0. Noting that $f(x)\leq m(x),$ we therefore have $|m(x+t)-m(x)|\leq max\{0,\,|max_{x\leq t\leq x+t}\,f(s)-f(x)|\}.$ The case t<0 is similar.2012-10-14