Difference between revisions of "Takagi function"
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Revision as of 17:13, 22 January 2016
Define the function $s(x)=\min_{n \in \mathbb{Z}} |x-n|$. The Takagi function (also called the blancmange function) is defined by $$\mathrm{takagi}(x)=\displaystyle\sum_{k=0}^{\infty} \dfrac{s(2^n x)}{2^n}.$$
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: █
