CONTINUOUS NOWHERE MONOTONIC FUNCTION, DEFINED BY TERMS CONTINUED A-REPRESENTATIONS OF NUMBERS

Abstract

We study structural and variational properties of one continued class of nowhere monotonic continuous functions unbounded variational, defined equality \[f(x=\Delta^{A_3}_{\alpha_1\alpha_2...\alpha_n...})=\Delta^{A_2}_{\beta_1\beta_2...\beta_n...},\] \[\beta_1=\begin{cases} 1 & \mbox{if } \alpha_1=2,\\ 0 & \mbox{if } \alpha_1\neq 2, \end{cases}\;\;\;\; \beta_{n+1}=\begin{cases} \beta_{n} & \mbox{if } \alpha_n+\alpha_{n+1}\neq 2,\\ 1-\beta_{n} & \mbox{if } \alpha_n+\alpha_{n+1}=2, \end{cases} \alpha_n \in \{0,1,2\}, n\in N,\] argument and values of which presented by form continued fraction. Elements $a_n$ of continued fraction $[0;a_1,a_2,...,a_n,...]$, consist to three- and two-symbol sets ($A_e=\{e_0,e_1,e_2\}$ $A_{\tau}=\{\tau_0,\tau_1\}$) corresponding. The function is analog of Bush-Wunderlich function and Tribin-function.