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Is there a real function that is differentiable at any point but nowhere monotone?

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    And see [here](http://math.stackexchange.com/a/802647/462) for details of the Katznelson-Stromberg construction.2014-05-21

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Yes. See for example "Everywhere Differentiable, Nowhere Monotone, Functions" by Y. Katznelson and Karl Stromberg.

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    The link above seems broken as well. Anyway, details have been posted [here](http://math.stackexchange.com/a/802647/462).2014-05-21