Difference between revisions of "Van der Waerden function"
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Revision as of 18:35, 24 May 2016
Define $s(x)=\inf_{n \in \mathbb{Z}} |x-n|$ (i.e. the distance from $x$ to the set of integers $\mathbb{Z}$). The van der Waerden function is defined by the formula $$V(x)=\displaystyle\sum_{k=0}^{\infty} \dfrac{s \left(10^k x \right)}{10^k}.$$ Note: to calculate $s(x)$ you may use $s(x)=\min \left(2^n x - \lfloor 2^n x \rfloor, \lceil 2^n x \rceil - x \right)$, where $\lfloor \cdot \rfloor$ denotes the floor function and $\lceil \cdot \rceil$ denotes the ceiling function.
Plot of the van der Waerden function.
Properties
Theorem: The van der Waerden function is continuous.
Proof: █
Theorem: The van der Waerden function is nowhere differentiable on $\mathbb{R}$.
Proof: █
