Difference between revisions of "Knopp function"
From specialfunctionswiki
(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 $$K_{a,b}(x)=\displaystyle\sum_{k=0}^{\infty} a...") |
(No difference)
|
Revision as of 12:09, 5 January 2016
Let $a \in (0,1), ab > 4,$ and $b>1$ an even integer. Define the Knopp function $K \colon \mathbb{R} \rightarrow \mathbb{R}$ by $$K_{a,b}(x)=\displaystyle\sum_{k=0}^{\infty} a^k \phi \left( b^k x \right),$$ where $\phi(x)=\displaystyle\inf_{m \in \mathbb{Z}} |x-m|$.
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: █
