Differentiable+Not monotone

Question

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

the constant function works, but I assume that example should be disallowed. — Sean Tilson, Jan 11 '11 at 14:58
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
See also the MO version of this question for additional details and references. — Andrés E. Caicedo, May 16 '14 at 16:34
And see here for details of the Katznelson-Stromberg construction. — Andrés E. Caicedo, May 21 '14 at 03:05

score 13 · Accepted Answer · edited Feb 16 '17 at 05:01

13

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

edited Feb 16 '17 at 05:01

Jonas Meyer

answered Jan 11 '11 at 07:48

Chris Eagle

2

The link doesn't work; is it just me? – Jonas Meyer Nov 20 '11 at 01:47
@Jonas: Google katznelson-stromberg.pdf and then click on "Quick view" on the first result. – Andrés E. Caicedo Nov 20 '11 at 02:03
The link above seems broken as well. Anyway, details have been posted here. – Andrés E. Caicedo May 21 '14 at 03:05

1 Answers1