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I realized I didn't know the answer to that question which seems simple, but I couldn't figure it out myself after reviewing my textbooks and draw a conclusion on my own. Suppose we have a function $f:I\to\mathbb{R}$, where $I$ is an interval of $\mathbb{R}$, $f$ is differentiable on $I$.

Is $f'$ automatically piecewise continuous ? That way it would have chance of being Riemann integrated.

Is there a pathological function, which can be differentiated, but its derivative is not piecewise continuous ? I know of the classic example $x\mapsto x^2\sin \frac{1}{x}$ which is differentiable everywhere but the derivative is discontinuous at $x=0$, but I can't think of a situation like this happening so much that you can't integrate $f'$.

Parfig
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Arno
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  • Let me introduce... the Volterra's function! – Sangchul Lee Mar 22 '25 at 23:33
  • Oh wow ! It's weird I also tried to google it, but I must have been so clumsy typing what I was trying to say in google that it didn't show me this result. Thank you, I guess that instantaneously answers my question ! Should I award you points for answering ? Thank you again. – Arno Mar 22 '25 at 23:38
  • @Surgeon Oh wow, that didn't pop up in the suggestions when I created my question. Thanks ! The Volterra function is mentioned, in one of the answers. Sorry for cluttering the space ! – Arno Mar 22 '25 at 23:40
  • @Arno No worries. I just happened to have seen that other post earlier and remembered it. I also just added a comment to that post which you may find interesting. – The Surgeon of Death Mar 22 '25 at 23:53
  • Ah yes, I've heard of the Henstock-Kurzweil integral before, but never strayed from my textbooks and always stuck to Riemann integrals while waiting to be graduate to Lebesgues integrals. I hear Henstock-Kurzweil integrals are very elegant and even rival Lebegues. I'll have to look sometime ! Currently though, my questions is broader than the one I posted: can $f$ be positive, monotonically decreasing, differentiable, and pathological like the Volterra function ? But I think I'm overthinking a silly problem I don't really need to solve. – Arno Mar 23 '25 at 00:03

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