Title: 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 this https URL). 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
this https URL). 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"
(this https URL) and in the several conference papers
(available at: this https URL,
this https URL).