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Is there a function whose derivative is discontinuous everywhere?

As is well-known by Darboux, a derivative $f'$ could have discontinuity of second type. My question is: whether there exists a function $f$, such that $f'$ is discontinuous everywhere.

Some Guy
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xldd
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    https://math.stackexchange.com/questions/112067/how-discontinuous-can-a-derivative-be?rq=1 This might help. – Literally an Orange Mar 26 '21 at 01:44
  • @LiterallyanOrange Very helpful! The post states that the derivative is continuous on a dense subset of its interval domain. – Theo Bendit Mar 26 '21 at 01:48
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    the reason that $f'$ must be continuous on an everywhere dense set is actually not that complicated - $f'$ is the pointwise limit of continuous functions or of first Baire class - one can take $f_n(x)=n(f(x+1/n)-f(x))$ at least for $f'$ restricted to a slightly smaller interval to the right and same to the left with $-1/n$ etc - and it is a standard manipulation of topological stuff to show that the points of discontinuity of Baire first-class functions are a countable union of nowhere dense $F_{\sigma}$ sets (functions in Baire first-class satisfy $f^{-1}(U)$ is $F_{\sigma}$ for open $U$) – Conrad Mar 26 '21 at 02:15

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