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How discontinuous can a derivative be?

$x^2\cos(1/x)$ is the standard example for a differentiable function whose derivative is not continuous at $x=0$.

But is there also a differentiable function $f:\Bbb R \rightarrow \Bbb R$ such that there is no $x_0 \in \Bbb R$ with $f'$ continuous at $x_0$?

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Dominik
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  • The answer is no. The continuity set of the derivative of an everywhere differentiable function is dense, and has cardinality of the continuum in any sub interval. The answer given in the post that the duplicate banner links to discusses this and much more. – Willie Wong Jan 28 '13 at 12:43

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