In this paper, we introduce and study a continuous function that projects the digits of the classical binary representation of an arbitrary real number onto the digits of the continued $A_2$-representation with zero redundancy. The function is defined by
\[f(\sum\limits_{n=1}^{\infty}\frac{\alpha_n}{2^n})=
1/2^{\alpha_1-1}+1/2^{-\alpha_2}+1/2^{\alpha_3-1}+1/2^{-\alpha_4}+...+
1/2^{\alpha_{2k-1}-1}+1/2^{-\alpha_{2k}}+..., \]
where $\alpha_n\in \{0;1\}$. We justify the correctness of this definition, which is nontrivial because some numbers possess two distinct binary expansions. Using the fact that almost all real numbers have normal binary expansions, together with Lebesgue's theorem on the derivative of continuous monotone functions, we show that the function $f$ is singular. Here, a singular function refers to a continuous, non-constant function whose derivative is equal to zero almost everywhere with respect to the Lebesgue measure.

In addition, we reveal the relationship between the function $f$, the right-shift digit operator, and inversor of digits of the $A_2$-representation of numbers. Based on this relationship, we also demonstrate the singularity of the inversor of digits of the $A_2$-representation.

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