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

TL;DR

This paper investigates a class of continuous functions with fractal properties, establishing criteria for their monotonicity, differentiability, and singularity, and analyzing their level sets.

Contribution

It introduces new criteria for monotonicity, non-monotonicity, and singularity in a class of nowhere monotonic fractal functions, expanding understanding of their properties.

Findings

Criteria for strict monotonicity and non-monotonicity established

Conditions for non-differentiability and singularity derived

Analysis of level set properties of the functions conducted

Abstract

We study one class of continuous functions $f$ defined on segment $[0,1]$ by equality $$ f(x)=\delta_{\alpha_1(x)1}+\sum^{\infty}_{k=2}\left[\delta_{\alpha_k(x)k}\prod^{k-1}_{j=1}g_{\alpha_j (x)j}\right]\equiv\Delta^{G^*_3}_{\alpha_1\alpha_2\ldots\alpha_k\ldots}, $$ where $||q^*_{ik}||$ is given infinite stochastic positive matrix ($i=0,1,2$; $k \in N$); $\beta_{0k}=0$, $\beta_{1k}=q_{0k}$, $\beta_{2k}=q_{0k}+q_{1k}$; $(\varepsilon_k)$ is given sequence of numbers such that $0\leqslant \varepsilon_k \leqslant 1 $; $g_{0k}=\dfrac{1+\varepsilon_k}{3}=g_{2k}$, $g_ {1k}=\dfrac{1-2\varepsilon_k}{3}$, $\delta_{0k}=0$, $\delta_{1k}=g_{0k}$, $\delta_{2k}=g_{0k}+g_{1k}$, $k\in N$. 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.