Takagi function

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Define the function $s(x)=\min_{n \in \mathbb{Z}} |x-n|$. The blancmange function (also called the Takagi function) is defined by $$\mathrm{blanc}(x)=\displaystyle\sum_{k=0}^{\infty} \dfrac{s(2^n x)}{2^n}.$$

Properties

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

Proof:

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

Proof:

References

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