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.

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?

I'msaying is that neither closure (in this sense) nor putting the question on hold would be appropriate here. $\endgroup$ – Todd Trimble♦ May 16 '14 at 14:104more comments