## Return to Answer

1

According to Weierstrass, Riemann knew about the existence of continuous nowhere differentiable functions. (Weierstrass' celebrated example was published in 1872, some 6 years after Riemann's death.) In his lectures, Riemann allegedly suggested the example $$f(x)=\sum\limits_{k=1}^{\infty}\frac{\sin k^2x}{k^2}$$ as early as 1861. He gave no proof and just mentioned that it could had been obtained from the theory of elliptic functions (see the historical note "Riemann’s example of a continuous “nondifferentiable” function continued" by S.L. Segal).

Hardy proved in 1916 that $f$ has no finite derivative at any $x=\pi\xi$ where $\xi$ is irrational but left the general case open.

It was only in 1970 that J. Gerver finally proved that the Riemann function is in fact differentiable when $$x=\pi\frac{2m+1}{2n+1},\qquad m,n\in\mathbb Z,$$ and $f'(x)=-1/2$ at these points ("The Differentiability of the Riemann Function at Certain Rational Multiples of π", ).