Difference between revisions of "Knopp function"

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Let $a \in (0,1), ab > 4,$ and $b>1$ an even integer. Define the Knopp function $K \colon \mathbb{R} \rightarrow \mathbb{R}$ by
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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=
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[[Knopp function is continuous]]<br />
<strong>Theorem:</strong> The [[Knopp function]] $K_{a,b}$ is [[continuous]] on $\mathbb{R}$ for $a \in (0,1)$ and $ab>1$.
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[[Knopp function is nowhere differentiable]]<br />
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<strong>Proof:</strong> █
 
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</div>
 
 
 
<div class="toccolours mw-collapsible mw-collapsed">
 
<strong>Theorem:</strong> The [[Knopp function]] $K_{a,b}$ is [[nowhere differentiable]] on $\mathbb{R}$ for $a \in (0,1)$and $ab > 1$.
 
<div class="mw-collapsible-content">
 
<strong>Proof:</strong> █
 
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</div>
 
  
 
=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

References

[1]