Difference between revisions of "Van der Waerden function"
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(Created page with "The van der Waerden function is defined by the formula $$V(x)=\displaystyle\sum_{k=0}^{\infty} \dfrac{1}{10^k} \underset{m\in\mathbb{Z}}{\inf} |10^k x-m|.$$ =Properties= <div...") |
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| − | The van der Waerden function is defined by the formula | + | The van der Waerden function $V \colon \mathbb{R} \rightarrow \mathbb{R}$ is defined by the formula |
| − | $$V(x)=\displaystyle\sum_{k=0}^{\infty} \dfrac{ | + | $$V(x)=\displaystyle\sum_{k=0}^{\infty} \dfrac{\mathrm{dist}_{\mathbb{Z}} \left(10^k x \right)}{10^k},$$ |
| + | where $\mathrm{dist}_{\mathbb{Z}}$ denotes the [[distance to integers]] function. | ||
| + | |||
| + | <div align="center"> | ||
| + | <gallery> | ||
| + | File:Vanderwaerdenplot.png|Plot of the van der Waerden function. | ||
| + | </gallery> | ||
| + | </div> | ||
=Properties= | =Properties= | ||
| − | + | [[van der Waerden function is continuous]] <br /> | |
| − | + | [[van der Waerden function is nowhere differentiable]]<br /> | |
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| − | </ | ||
| − | + | =See Also= | |
| − | + | [[Takagi function]] | |
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=References= | =References= | ||
[https://pure.ltu.se/ws/files/30923977/LTU-EX-03320-SE.pdf] <br /> | [https://pure.ltu.se/ws/files/30923977/LTU-EX-03320-SE.pdf] <br /> | ||
| + | |||
| + | {{:Continuous nowhere differentiable functions footer}} | ||
| + | |||
| + | [[Category:SpecialFunction]] | ||
Latest revision as of 03:33, 6 July 2016
The van der Waerden function $V \colon \mathbb{R} \rightarrow \mathbb{R}$ is defined by the formula $$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.
Plot of the van der Waerden function.
Properties
van der Waerden function is continuous
van der Waerden function is nowhere differentiable
