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.

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