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The fact that $f(x)$ is differetiable dosen't guarantee the integrability of $f'(x)$ on $[a,b]$ beacuse $f'(x)$ could be unbounded, which violates the assumption of Riemann integral. However, what's the case if $f'(x)$ is bounded? Since $f'(x)$ doesn't have any discontinuity point of the first kind and $f'(x)$ is bounded, $f'(x)$ has only discontinuity points as $\sin \frac1x$ dose near $x=0$ if there are any.

Could $f'(x)$ has uncountably many discontinuity points of that kind? Otherwise, $f'(x)$ should be integrable on $[a,b]$. I think $f'(x)$ is integrable if it is bounded, could someone please give me a hand here? Or is there a counterexample?

Thanks in advance!

Steve
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1 Answers1

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Yes, there are differentiable functions $f$ such that $f'$ is bounded but not Riemann-integrable. The standard example is Volterra's function.

José Carlos Santos
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