In the paper we study cardinality of the set $P_c$ of continuous on $[0,1]$ functions preserving digit 1 in three-symbol self-similar $Q_3$-representation of a number. This representation generalizes the classic ternary representation: $x=\sum\limits_{k=1}^{\infty} 3^{-k}\alpha_k(x) \equiv\Delta^{3}_{\alpha_1 \alpha_2 \ldots \alpha_n \ldots}$, where $\alpha_n(x) \in A_3\equiv\{0,1,2\}$.

All functions from class $P_c$ are given in the following form:

$y = f(x) = f(\Delta^{Q_3}_{\alpha_1(x) \alpha_2(x) \ldots \alpha_n(x) \ldots}) = \Delta^{Q_3}_{γ_1 γ_2 \ldots γ_n \ldots},$ where $a_n ∈ A_3, γ_n ∈ A_3,$

$x = \Delta^{Q_3}_{\alpha_1(x) \alpha_2(x) \ldots \alpha_n(x) \ldots} ≡ β_{\alpha_1(x)} + \sum\limits_{k=2}^∞[β_{\alpha_k(x)}\prod\limits_{j =1}^{k-1} q_{\alpha_j(x)}]$

and $\gamma_n= \gamma_n(\alpha_1(x), \alpha_2(x), \ldots, \alpha_n(x))$ but $\gamma_n =1$ if and only if $\alpha_n(x) =1$.
We prove that $P_c$ is a continuum set. Analytical expression for functions from class $P_c$ are obtained. Their variational and integral properties are also studied.

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