Van der Waerden function

The van der Waerden function $V \colon \mathbb{R} \rightarrow \mathbb{R}$ is defined by the formula $$V(x)=\displaystyle\sum_{k=0}^{\infty} \dfrac{\mathrm{dist}_{\mathbb{Z}} \left(10^k x \right)}{10^k},$$ where $\mathrm{dist}_{\mathbb{Z}}$ denotes the distance to integers function.

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  Plot of the van der Waerden function.

Properties

van der Waerden function is continuous
van der Waerden function is nowhere differentiable

See Also

Takagi function

References

[1]

Continuous nowhere differentiable functions

Darboux function

Faber F1

Faber F2

Takagi

van der Waerden