Takagi function

The Takagi function (also called the blancmange function) is defined by $$\mathrm{takagi}(x)=\displaystyle\sum_{k=0}^{\infty} \dfrac{\mathrm{dist}_{\mathbb{Z}}(2^n x)}{2^n}.$$

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  Graph of $\mathrm{takagi}$ on $[0,1]$.

Properties

Theorem: The Takagi function is continuous on $\mathbb{R}$.

Proof: █

Theorem: The Takagi function is nowhere differentiable on $\mathbb{R}$.

Proof: █

See Also

van der Waerden function

References

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