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

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the constant function works, but I assume that example should be disallowed. –  Sean Tilson Jan 11 '11 at 14:58
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Doesn't "monotone" by default mean weakly, not strictly, monotone? (i.e. monotone increasing means for all $x$, $y$, $x \le y \Rightarrow f(x) \le f(y)$). So constant functions are everywhere monotone. –  Chris Eagle Jan 11 '11 at 18:33
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See also the MO version of this question for additional details and references. –  Andres Caicedo May 16 at 16:34
    
And see here for details of the Katznelson-Stromberg construction. –  Andres Caicedo May 21 at 3:05

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up vote 11 down vote accepted

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

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The link doesn't work; is it just me? –  Jonas Meyer Nov 20 '11 at 1:47
    
@Jonas: Google katznelson-stromberg.pdf and then click on "Quick view" on the first result. –  Andres Caicedo Nov 20 '11 at 2:03
    
@Andres: Thank you. –  Jonas Meyer Nov 20 '11 at 2:08
    
The link above seems broken as well. Anyway, details have been posted here. –  Andres Caicedo May 21 at 3:05

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