Difference between revisions of "Takagi function"
From specialfunctionswiki
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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 | Define the function $s(x)=\min_{n \in \mathbb{Z}} |x-n|$. The blancmange function (also called the Takagi function) is defined by | ||
$$\mathrm{blanc}(x)=\displaystyle\sum_{k=0}^{\infty} \dfrac{s(2^n x)}{2^n}.$$ | $$\mathrm{blanc}(x)=\displaystyle\sum_{k=0}^{\infty} \dfrac{s(2^n x)}{2^n}.$$ | ||
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<div align="center"> | <div align="center"> | ||
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</gallery> | </gallery> | ||
</div> | </div> | ||
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| + | =Properties= | ||
| + | <div class="toccolours mw-collapsible mw-collapsed"> | ||
| + | <strong>Theorem:</strong> The blancmange function is [[continuous]] on $\mathbb{R}$. | ||
| + | <div class="mw-collapsible-content"> | ||
| + | <strong>Proof:</strong> █ | ||
| + | </div> | ||
| + | </div> | ||
| + | |||
| + | <div class="toccolours mw-collapsible mw-collapsed"> | ||
| + | <strong>Theorem:</strong> The Blancmange function is [[nowhere differentiable]] on $\mathbb{R}$. | ||
| + | <div class="mw-collapsible-content"> | ||
| + | <strong>Proof:</strong> █ | ||
| + | </div> | ||
| + | </div> | ||
| + | |||
| + | =References= | ||
| + | [https://pure.ltu.se/ws/files/30923977/LTU-EX-03320-SE.pdf] | ||
Revision as of 23:03, 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: █
