On one nearly everywhere continuous and nowhere differentiable function, that defined by automaton with finite memory

Abstract

This paper is devoted to the investigation of the following function $$ f: x=\Delta^{3}_{\alpha_{1}\alpha_{2}...\alpha_{n}...}{\rightarrow} \Delta^{3}_{\varphi(\alpha_{1})\varphi(\alpha_{2})...\varphi(\alpha_{n})...}=f(x)=y, $$ where $\varphi(i)=\frac{-3i^{2}+7i}{2}$, $ i \in N^{0}_{2}=\{0,1,2\}$, and $\Delta^{3}_{\alpha_{1}\alpha_{2}...\alpha_{n}...}$ is the ternary representation of $x \in [0;1]$. That is values of this function are obtained from the ternary representation of the argument by the following change of digits: 0 by 0, 1 by 2, and 2 by 1. This function preserves the ternary digit $0$. Main mapping properties and differential, integral, fractal properties of the function are studied. Equivalent representations by additionally defined auxiliary functions of this function are proved. This paper is the paper translated from Ukrainian (the Ukrainian variant available at https://www.researchgate.net/publication/292970012). In 2012, the Ukrainian variant of this paper was represented by the author in the International Scientific Conference "Asymptotic Methods in the Theory of Differential Equations" dedicated to 80th anniversary of M. I. Shkil (the conference paper available at https://www.researchgate.net/publication/301765319). In 2013, the investigations of the present article were generalized by the author in the paper "One one class of functions with complicated local structure" (https://arxiv.org/pdf/1601.06126.pdf) and in the several conference papers (available at: https://www.researchgate.net/publication/301765326, https://www.researchgate.net/publication/303052308).