I would like to construct a bounded, real function $f:[a,b] \to \mathbb{R}$, which is not Riemann integrable, but has antiderivatives. I can easily construct an unbounded function with this property, but finding a bounded one is significantly more difficult.
All I could find was this topic (Non-integrable function that has an antiderivative) which offers some valuable insight into this matter, but doesn't actually construct a function with this property.
I also wonder if such a function can be expressed in terms of elementary functions because in the topic above some results from measure theory are mentioned and these are beyond my reach.
Thank you for taking your time to read my question !

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    $\begingroup$ See this. $\endgroup$ – David Mitra Aug 12 at 19:19
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    $\begingroup$ Also, this. $\endgroup$ – David Mitra Aug 12 at 19:25
  • $\begingroup$ "integrable" mean Riemann integrable? $\endgroup$ – zhw. Aug 12 at 19:58
  • $\begingroup$ @zhw Yes, it does, sorry for the ambiguity, I will edit it. $\endgroup$ – Alexdanut Aug 12 at 21:04

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