Difference between revisions of "Knopp function"

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=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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<strong>Theorem:</strong> The [[Knopp function]] $K_{a,b}$ is [[nowhere differentiable]] on $\mathbb{R}$ for $a \in (0,1)$and $ab > 1$.
 
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<strong>Proof:</strong> █
 
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=See Also=
 
=See Also=

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]