Knopp function

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: █

References

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