Difference between revisions of "Takagi function"
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| − | Define the function $s(x)=\min_{n \in \mathbb{Z}} |x-n|$. The | + | Define the function $s(x)=\min_{n \in \mathbb{Z}} |x-n|$. The Takagi function (also called the blancmange function) is defined by |
| − | $$\mathrm{ | + | $$\mathrm{takagi}(x)=\displaystyle\sum_{k=0}^{\infty} \dfrac{s(2^n x)}{2^n}.$$ |
<div align="center"> | <div align="center"> | ||
<gallery> | <gallery> | ||
| − | File:Blancmangefunction.png|Graph of $\mathrm{ | + | File:Blancmangefunction.png|Graph of $\mathrm{takagi}$ on $[0,1]$. |
</gallery> | </gallery> | ||
</div> | </div> | ||
| Line 10: | Line 10: | ||
=Properties= | =Properties= | ||
<div class="toccolours mw-collapsible mw-collapsed"> | <div class="toccolours mw-collapsible mw-collapsed"> | ||
| − | <strong>Theorem:</strong> The | + | <strong>Theorem:</strong> The Takagi function is [[continuous]] on $\mathbb{R}$. |
<div class="mw-collapsible-content"> | <div class="mw-collapsible-content"> | ||
<strong>Proof:</strong> █ | <strong>Proof:</strong> █ | ||
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<div class="toccolours mw-collapsible mw-collapsed"> | <div class="toccolours mw-collapsible mw-collapsed"> | ||
| − | <strong>Theorem:</strong> The | + | <strong>Theorem:</strong> The Takagi function is [[nowhere differentiable]] on $\mathbb{R}$. |
<div class="mw-collapsible-content"> | <div class="mw-collapsible-content"> | ||
<strong>Proof:</strong> █ | <strong>Proof:</strong> █ | ||
Revision as of 16:51, 22 January 2016
Define the function $s(x)=\min_{n \in \mathbb{Z}} |x-n|$. The Takagi function (also called the blancmange function) is defined by $$\mathrm{takagi}(x)=\displaystyle\sum_{k=0}^{\infty} \dfrac{s(2^n x)}{2^n}.$$
- Blancmangefunction.png
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: █
