It is well known that there are functions $f \colon \mathbb{R} \to \mathbb{R}$ that are everywhere continuous but nowhere monotonic (i.e. the restriction of $f$ to any non-trivial interval $[a,b]$ is not monotonic), for example the [Weierstrass function](http://en.wikipedia.org/wiki/Weierstrass_function).


It’s easy to prove that there are no such functions if we add the condition that $f$ is continuously differentiable, so it is natural to ask the same question, with $f$ only differentiable. This seems to me a non trivial question, since, at least a priori, the derivative $f'$ could change sign on any non trivial interval, so we cannot use the standard results to prove the monotonicity of $f$.



> 
> Question: Does it exist a function $f \colon \mathbb{R} \to \mathbb{R}$ that is everywhere differentiable but nowhere monotonic?
> 
> 
> 



Everywhere differentiable function that is nowhere monotonic
Everywhere differentiable but nowhere monotonic real functions do exist. It seems that the first correct examples were found by A. Denjoy in [this paper](http://www.numdam.org/numdam-bin/item?id=BSMF_1915__43__161_0). A short existence proof, based on Baire's category theorem, was given by C. E. Weil in [this paper](http://www.ams.org/journals/proc/1976-056-01/S0002-9939-1976-0396870-2/).


