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What example can be given for a $f\colon [0,1]\to \mathbb R$, differentiable at all points of the segment $[0,1]$ and such that $f'$ is not Riemann integrable by $[0,1]$?

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    What are your thoughts on the question ? – TheSilverDoe Dec 25 '21 at 18:21
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    It's not exactly trivial: https://en.m.wikipedia.org/wiki/Volterra%27s_function – Vercassivelaunos Dec 25 '21 at 18:32
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    See this. – David Mitra Dec 25 '21 at 18:38
  • My thoughts is only to use Lebesgue criteria, but I don't know how to exactly –  Dec 25 '21 at 18:52
  • The link to the Volterra function is the right advice. The key to the construction is to know that there is a fat Cantor set (i.e., a perfect, nowhere dense subset $P$ of $[0,1]$ that does not have measure zero). The idea is to set $f(x)=0$ on $P$ and adjust the "humps" on the complementary intervals so that $f'$ is discontinuous at points of $P$. I don't see this as a homework problem, more something worth researching. – B. S. Thomson Dec 26 '21 at 00:35

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