Everywhere differentiable function that is nowhere monotonic 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?

 A: Everywhere differentiable but nowhere monotonic real functions do exist. It seems that the first correct examples were found by A. Denjoy in this paper. A short existence proof, based on Baire's category theorem, was given by C. E. Weil in this paper.
A: A good reference for this is the paper 

Y. Katznelson and K. Stromberg. "Everywhere differentiable, nowhere monotone, functions, Am. Math. Monthly, 81 (4), (1974), 349-353.

There, the authors give an explicit construction of a function $L:\mathbb R\to\mathbb R$ that is differentiable, with bounded derivative, and such that all three sets $\{x\mid L'(x)>0\}$, $\{x\mid L'(x)=0\}$, and $\{x\mid L'(x)<0\}$ are dense in $\mathbb R$. The argument is flexible enough that we can arrange $\{x\mid L'(x)=0\}=\mathbb Q$, if wanted. The function $L$ is the difference of two monotone functions, and $L'$ is not Riemannn integrable over any interval. (Details of the construction can be found here.)
In the paper, they indicate that the first example of a differentiable, nowhere monotone, function is due to Köpke, in 1889, with a further example due to Pereno in 1897. (I have not examined their constructions myself, so I do not know whether there are mistakes in their presentation, as the other answer suggests, but Pereno's function is discussed as an example in Hobson's book from 1957.) 

For this kind of results, two obvious references to consult are Counterexamples in Analysis, by Gelbaum and Olmsted, and A second course on real functions, by van Rooij and Schikhof. The latter presents the Katznelson-Stromberg construction in detail.
A: There is a curious closed subspace of the Banach space of bounded functions on $\mathbb R$ : the space of bounded functions that vanish in a dense set, and are derivatives of everywhere differentiable functions. An easy application of the Baire category theorem, due to Weil, shows that the subset of functions that change sign on every interval is of second category in this Banach space. Their antiderivative is, of course, nowhere monotonic.
