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Is there a function $f$: $\mathbb R\to \mathbb R$ such that $f$ is differentiable everywhere but $f'$ is discontinuous everywhere?


marked as duplicate by user147263, GEdgar, Johanna, dustin, Daniel W. Farlow Feb 27 '15 at 2:48

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  • $\begingroup$ Volterra's function is discontinuous in a set of positive measure. $\endgroup$ – Jorge Fernández Hidalgo Feb 27 '15 at 2:02

No, $f'$ is of first Baire class (a pointwise limit of continuous functions), so it is continuous everywhere except a meager (first category) set.

$f$ is differentiable, so $f$ is continuous, and $$ f'(x) = \lim_{n \to \infty} g_n(x),\qquad\text{where}\qquad g_n(x) = \frac{f(x+1/n)-f(x)}{1/n} $$ and $g_n$ is continuous.


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