On one class of nowhere non-monotonic functions with fractal properties that contains a subclass of singular functions

Abstract

We study one class of continuous functions $f$ defined on segment $[0,1]$ by equality $$f(x)={\delta }_{{\alpha }_1(x)1}+\sum _{k=2}^{\infty }\left[{\delta }_{{\alpha }_k(x)k}\prod _{j=1}^{k-1}{g}_{{\alpha }_j(x)j}\right]\equiv {\Delta }_{{\alpha }_1{\alpha }_2\dots {\alpha }_k\dots }^{G_3^{\ast }},$$ where $||q_{ik}^{\ast }||$ is given infinite stochastic positive matrix ($i=0,1,2$; $k\in N$); ${\beta }_{0k}=0$, ${\beta }_{1k}=q_{0k}$, ${\beta }_{2k}=q_{0k}+q_{1k}$; $({\epsilon }_k)$ is given sequence of numbers such that $0⩽{\epsilon }_k⩽1$; $g_{0k}=\frac{1+{\epsilon }_k}{3}=g_{2k}$, $g_{1k}=\frac{1-2{\epsilon }_k}{3}$, ${\delta }_{0k}=0$, ${\delta }_{1k}=g_{0k}$, ${\delta }_{2k}=g_{0k}+g_{1k}$, $k\in N$. We found criteria of strict monotonicity, non monotonicity and nowhere monotonicity, non-differentiability and singularity of the functions. We pay attention to properties of level sets of the functions.