Difference between revisions of "Takagi function"
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Revision as of 23:07, 31 December 2015
Define the function $s(x)=\min_{n \in \mathbb{Z}} |x-n|$. The blancmange function (also called the Takagi function) is defined by $$\mathrm{blanc}(x)=\displaystyle\sum_{k=0}^{\infty} \dfrac{s(2^n x)}{2^n}.$$
- Blancmangefunction.png
Graph of $\mathrm{blanc}$ on $[0,1]$.
Properties
Theorem: The blancmange function is continuous on $\mathbb{R}$.
Proof: █
Theorem: The Blancmange function is nowhere differentiable on $\mathbb{R}$.
Proof: █
