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 $$ f(x)=δ_{α_1(x)1}+\sum^{\infty}_{k=2}\left[δ_{α_k(x)k}\prod^{k-1}_{j=1}g_{α_j (x)j}\right]\equivΔ^{G^*_3}_{α_1α_2\ldotsα_k\ldots}, $$ where $||q^*_{ik}||$ is given infinite stochastic positive matrix ($i=0,1,2$; $k \in N$); $β_{0k}=0$, $β_{1k}=q_{0k}$, $β_{2k}=q_{0k}+q_{1k}$; $(\varepsilon_k)$ is given sequence of numbers such that $0\leqslant \varepsilon_k \leqslant 1 $; $g_{0k}=\dfrac{1+\varepsilon_k}{3}=g_{2k}$, $g_ {1k}=\dfrac{1-2\varepsilon_k}{3}$, $δ_{0k}=0$, $δ_{1k}=g_{0k}$, $δ_{2k}=g_{0k}+g_{1k}$, $k\in N$. 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.

Theorems & Definitions (15)

• Lemma 1
• proof
• Theorem 1
• proof
• Corollary 1
• Theorem 2
• proof
• Corollary 2
• Corollary 3
• Theorem 3