There are plenty of examples of differentiable functions $\Bbb R\to\Bbb R$ with derivatives that are not everywhere continuous. However, as stated here, it is impossible for the derivative to be nowhere continuous. In general, can anything be said about exactly how "ugly" a derivative can get?
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3See this. – David Mitra May 31 '14 at 10:10
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2There is the question: What is the measure of the "ugliness" of the derivative? Is it how many places it is not continuous? – FundThmCalculus May 31 '14 at 10:10