A unified approach for nondifferentiable functions (Q1323200)

Let \(\Psi(x)\) denote the distance between \(x\) and the nearest integer. The Knopps example \(K(x)= \sum^ \infty_{n=0} a^ n \Psi(b^ n x)\) for \(0< a< 1\), \(b\in \mathbb{N}\) and \(ab>4\) is a continuous non-differentiable function at any point. The authors study the function \(f(x)= \sum^ \infty_{n=0} a^ n \Psi(b^ n x+ \theta_ n)\) under the weaker condition \(ab>1\) for any sequence \(\theta_ n\). They show that \(f(x)\) has no right (left) derivative at any point \(x\) and it belongs to the Lipschitz class \(\alpha\) for \(\alpha=- \log a/\log b\).