Difference between revisions of "Knopp function"
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| − | Let $a \in (0,1) | + | Let $a \in (0,1)$ $ab > 1$. Define the Knopp function $K \colon \mathbb{R} \rightarrow \mathbb{R}$ by |
$$K_{a,b}(x)=\displaystyle\sum_{k=0}^{\infty} a^k \mathrm{dist}_{\mathbb{Z}} \left( b^k x \right),$$ | $$K_{a,b}(x)=\displaystyle\sum_{k=0}^{\infty} a^k \mathrm{dist}_{\mathbb{Z}} \left( b^k x \right),$$ | ||
where $\mathrm{dist}_{\mathbb{Z}}$ denotes the [[distance to integers]] function. | where $\mathrm{dist}_{\mathbb{Z}}$ denotes the [[distance to integers]] function. | ||
=Properties= | =Properties= | ||
| − | + | [[Knopp function is continuous]]<br /> | |
| − | + | [[Knopp function is nowhere differentiable]]<br /> | |
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=See Also= | =See Also= | ||
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=References= | =References= | ||
[https://pure.ltu.se/ws/files/30923977/LTU-EX-03320-SE.pdf] | [https://pure.ltu.se/ws/files/30923977/LTU-EX-03320-SE.pdf] | ||
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| + | [[Category:SpecialFunction]] | ||
Latest revision as of 03:31, 27 October 2016
Let $a \in (0,1)$ $ab > 1$. Define the Knopp function $K \colon \mathbb{R} \rightarrow \mathbb{R}$ by $$K_{a,b}(x)=\displaystyle\sum_{k=0}^{\infty} a^k \mathrm{dist}_{\mathbb{Z}} \left( b^k x \right),$$ where $\mathrm{dist}_{\mathbb{Z}}$ denotes the distance to integers function.
Properties
Knopp function is continuous
Knopp function is nowhere differentiable
See Also
Takagi function
van der Waerden function
