TITLE: Let $f:[0,1]\to [-1,1]$ be arbitrary and define $g(x)=\sup\limits_{a\leq t\leq x}|f(t)|.$ Must $g$ be Riemann integrable?
QUESTION [2 upvotes]: Let $f:[0,1]\to [-1,1]$ be arbitrary and define $g(x)=\sup\limits_{a\leq t\leq x}|f(t)|.$ Must $g$ be Riemann integrable? 
I'm of the feeling that $g$ is not Riemann integrable but I can't find a concrete example. Any help out there?

REPLY [4 votes]: The function $g$ is an increasing function. And a monotonic function defined on a closed and bounded interval is always Riemann-integrable. For instance, it has only countably many discontinuity points and therefore the set of discontinuity points has Lebesgue measure equal to $0$.
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