In the paper we consider the class of continuous on $[0;1]$ functions preserving digit 1 in three-symbol $Q_3$  -representation of a number. This representation generalizes the classical ternary representation: $x = \sum_{k=1}^∞ 3^{-k} a_k(x) ≡ \Delta_{α_1α_2...α_n...}^3,$ where $a_n(x) ∈ A_3 ≡ \{0,1,2\}.$ Namely we study functions of the form

$f(\Delta_{α_1α_2...α_n...}^{Q_3}) = \Delta_{γ_1γ_2...γ_n...}^{Q_3},$ where $α_n, γ_n ∈ A_3,$

$\Delta_{α_1α_2...α_n...}^{Q_3} ≡ β_{α_1(x)} + \sum_{k=2}^∞  [   β_{α_k(x)} \prod_{j=1}^{k-1} q_{α_j(x)}   ]$

and $γ_n = γ_n (α_1(x),α_2(x),...,α_n(x)),$ but $γ_n = 1$ if and only if $α_n = 1.$

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