Is there functions $f:[a,b]\longrightarrow \mathbb R$ derivable on $]a,b[$ s.t. $f'$ is almost nowhere continuous ? Or at least not continuous on a set of measure strictly positive.
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See How discontinuous can a derivative be? – Dave L. Renfro Nov 14 '17 at 11:27
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You have Volterrra's function, made by gluing together lots of copies of $x^2\sin(1/x)$. It has derivative which is discontinuous on a fat Cantor set, with positive measure.
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