CONTINUOUS NOWHERE MONOTONIC FUNCTION DEFINED IT TERM CONTINUED A_2-FRACTIONS REPRESENTATION OF NUMBERS

We consider finite class of functions defined by parameters $e_0,e_1,e_2$ belonging to the set $A=\{0,1\}$. The digits of the continued fraction $A_2$-representation of the argument $$x=\frac{1}{\alpha_1+\frac{1}{\alpha_2+_{\ddots}}}\equiv \Delta^A_{a_1...a_n...},$$ where $\alpha_n\in \{\frac{1}{2};1\}$, $a_n=2\alpha_n-1$, $n\in N$, and the values of the function are in a recursive dependence, namely: $$f(x=\Delta^A_{a_1...a_{2n}...})=\Delta^A_{b_1b_2...b_n...},$$ \begin{equation*} b_1=\begin{cases} e_0 &\mbox{ if } (a_1,a_2)=(e_1,e_2),\\ 1-e_0 &\mbox{ if } (a_1,a_2)\neq(e_1,e_2), \end{cases} \end{equation*} \begin{equation*} b_{k+1}=\begin{cases} b_k &\mbox{ if } (a_{2k+1},a_{2k+2})\neq(a_{2k-1},a_{2k}),\\ 1-b_k &\mbox{ if } (a_{2k+1},a_{2k+2})=(a_{2k-1},a_{2k}). \end{cases} \end{equation*} In the article, we justify the well-defined of the function, continuous and nowhere monotonic function. The variational properties of the function were studied and the unbounded variation was proved.