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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?

user586973
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1 Answers1

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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 $0$.

José Carlos Santos
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    To the asker: Riemann integrability of monotone function can be proved directly using the criteria based on difference of upper and lower Darboux sums. This avoids the slightly more complicated route based on Lebesgue's theorem mentioned in the answer. – Paramanand Singh Sep 07 '18 at 15:20
  • @José Carlos Santos: I have not yet done Lebesgue measure. Please, can you give me an example that suits Riemann integration? – user586973 Sep 08 '18 at 05:43
  • @Micheal You will find here a proof of the fact that monotonic $\implies$ Riemann-integrable. – José Carlos Santos Sep 08 '18 at 06:38