Riemann function

The Riemann function is the function $R \colon \mathbb{R} \rightarrow \mathbb{R}$ defined by $$R(x)=\displaystyle\sum_{k=1}^{\infty} \dfrac{\sin(k^2 x)}{k^2}.$$

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  Plot of $R(x)$ on $[0,1]$.

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  The partial sum $R(x,N)=\displaystyle\sum_{k=1}^N \dfrac{\sin(k^2 x)}{k^2}$ for various values of $N$.

Properties

Riemann function is continuous
Riemann function is almost nowhere differentiable

References

[1]