Is there a real function that is differentiable at any point but nowhere monotone?
Differentiable+Not monotone
10
$\begingroup$
real-analysis
-
0the constant function works, but I assume that example should be disallowed. – 2011-01-11
-
4Doesn'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. – 2011-01-11
-
2See also the [MO version](http://mathoverflow.net/q/167323/6085) of this question for additional details and references. – 2014-05-16
-
0And see [here](http://math.stackexchange.com/a/802647/462) for details of the Katznelson-Stromberg construction. – 2014-05-21
1 Answers
12
Yes. See for example "Everywhere Differentiable, Nowhere Monotone, Functions" by Y. Katznelson and Karl Stromberg.
-
2The link doesn't work; is it just me? – 2011-11-20
-
0@Jonas: Google katznelson-stromberg.pdf and then click on "Quick view" on the first result. – 2011-11-20
-
0The link above seems broken as well. Anyway, details have been posted [here](http://math.stackexchange.com/a/802647/462). – 2014-05-21