Difference between revisions of "Knopp function"
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(Created page with "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 $$K_{a,b}(x)=\displaystyle\sum_{k=0}^{\infty} a...") |
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Let $a \in (0,1), ab > 4,$ and $b>1$ an even integer. Define the Knopp function $K \colon \mathbb{R} \rightarrow \mathbb{R}$ by | Let $a \in (0,1), ab > 4,$ and $b>1$ an even integer. Define the Knopp function $K \colon \mathbb{R} \rightarrow \mathbb{R}$ by | ||
| − | $$K_{a,b}(x)=\displaystyle\sum_{k=0}^{\infty} a^k \ | + | $$K_{a,b}(x)=\displaystyle\sum_{k=0}^{\infty} a^k \mathrm{dist}_{\mathbb{Z}} \left( b^k x \right),$$ |
| − | where $\ | + | where $\mathrm{dist}_{\mathbb{Z}}$ denotes the [[distance to integers]] function. |
=Properties= | =Properties= | ||
Revision as of 19:45, 22 January 2016
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 $$K_{a,b}(x)=\displaystyle\sum_{k=0}^{\infty} a^k \mathrm{dist}_{\mathbb{Z}} \left( b^k x \right),$$ where $\mathrm{dist}_{\mathbb{Z}}$ denotes the distance to integers function.
Properties
Theorem: The Knopp function $K_{a,b}$ is continuous on $\mathbb{R}$ for $a \in (0,1)$ and $ab>1$.
Proof: █
Theorem: The Knopp function $K_{a,b}$ is nowhere differentiable on $\mathbb{R}$ for $a \in (0,1)$and $ab > 1$.
Proof: █
