Difference between revisions of "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
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The Takagi function (also called the blancmange function) is defined by
$$\mathrm{blanc}(x)=\displaystyle\sum_{k=0}^{\infty} \dfrac{s(2^n x)}{2^n}.$$
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$$\mathrm{takagi}(x)=\displaystyle\sum_{k=0}^{\infty} \dfrac{\mathrm{dist}_{\mathbb{Z}}(2^n x)}{2^n},$$
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where $\mathrm{dist}_{\mathbb{Z}}$ denotes the [[distance to integers]] function.
  
 
<div align="center">
 
<div align="center">
 
<gallery>
 
<gallery>
File:Blancmangefunction.png|Graph of $\mathrm{blanc}$ on $[0,1]$.
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File:Takagiplot.png|Graph of $\mathrm{takagi}$ on $[0,1]$.
 
</gallery>
 
</gallery>
 
</div>
 
</div>
  
 
=Properties=
 
=Properties=
<div class="toccolours mw-collapsible mw-collapsed">
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[[Takagi function is continuous]]<br />
<strong>Theorem:</strong> The blancmange function is [[continuous]] on $\mathbb{R}$.
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[[Takagi function is nowhere differentiable]]<br />
<div class="mw-collapsible-content">
 
<strong>Proof:</strong> █
 
</div>
 
</div>
 
  
<div class="toccolours mw-collapsible mw-collapsed">
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=See Also=
<strong>Theorem:</strong> The Blancmange function is [[nowhere differentiable]] on $\mathbb{R}$.
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[[van der Waerden function]]
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<strong>Proof:</strong> █
 
</div>
 
</div>
 
  
 
=References=
 
=References=
[https://pure.ltu.se/ws/files/30923977/LTU-EX-03320-SE.pdf]
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[https://pure.ltu.se/ws/files/30923977/LTU-EX-03320-SE.pdf]<br />
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[http://www.math.tamu.edu/~tvogel/gallery/node7.html]<br />
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{{:Continuous nowhere differentiable functions footer}}
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[[Category:SpecialFunction]]

Latest revision as of 03:33, 6 July 2016

The Takagi function (also called the blancmange function) is defined by $$\mathrm{takagi}(x)=\displaystyle\sum_{k=0}^{\infty} \dfrac{\mathrm{dist}_{\mathbb{Z}}(2^n x)}{2^n},$$ where $\mathrm{dist}_{\mathbb{Z}}$ denotes the distance to integers function.

Properties

Takagi function is continuous
Takagi function is nowhere differentiable

See Also

van der Waerden function

References

[1]
[2]

Continuous nowhere differentiable functions