Difference between revisions of "Van der Waerden function"

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The van der Waerden function is defined by the formula
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The van der Waerden function $V \colon \mathbb{R} \rightarrow \mathbb{R}$ is defined by the formula
$$V(x)=\displaystyle\sum_{k=0}^{\infty} \dfrac{1}{10^k} \underset{m\in\mathbb{Z}}{\inf} |10^k x-m|.$$
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$$V(x)=\displaystyle\sum_{k=0}^{\infty} \dfrac{\mathrm{dist}_{\mathbb{Z}} \left(10^k x \right)}{10^k},$$
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where $\mathrm{dist}_{\mathbb{Z}}$ denotes the [[distance to integers]] function.
  
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<div align="center">
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<gallery>
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File:Vanderwaerdenplot.png|Plot of the van der Waerden function.
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</gallery>
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</div>
 
=Properties=
 
=Properties=
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[[van der Waerden function is continuous]] <br />
<strong>Theorem:</strong> The van der Waerden function is [[continuous]].
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[[van der Waerden function is nowhere differentiable]]<br />
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<strong>Proof:</strong> █
 
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=See Also=
<strong>Theorem:</strong> The van der Waerden function is [[nowhere differentiable]] on $\mathbb{R}$.
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[[Takagi function]]
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<strong>Proof:</strong> █
 
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=References=
 
=References=
 
[https://pure.ltu.se/ws/files/30923977/LTU-EX-03320-SE.pdf] <br />
 
[https://pure.ltu.se/ws/files/30923977/LTU-EX-03320-SE.pdf] <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 van der Waerden function $V \colon \mathbb{R} \rightarrow \mathbb{R}$ is defined by the formula $$V(x)=\displaystyle\sum_{k=0}^{\infty} \dfrac{\mathrm{dist}_{\mathbb{Z}} \left(10^k x \right)}{10^k},$$ where $\mathrm{dist}_{\mathbb{Z}}$ denotes the distance to integers function.


Properties

van der Waerden function is continuous
van der Waerden function is nowhere differentiable

See Also

Takagi function

References

[1]

Continuous nowhere differentiable functions