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", ).

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", ).

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", ).

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", ).