In the paper we consider a continuum class functions defined by terms of the $Q_s$-representation of real numbers on the segment $[0;1]$, which generalizes the classical $s$-adic representation. The dependence of the $n$-th digit of the $Q_s$-representation of the function value is specified by a finite function $\varphi_n(a_n,\alpha_n)$ of two variables, whose arguments are the corresponding $Q_s$-digits $\alpha_n(x)$ and $a_n(a)$ of the input
$x$ and the parameter $a$, respectively.

We prove continuity of each function in this class at every $Q_s$-unary number, i.e., at points possessing a unique $Q_s$-representation. Necessary and sufficient conditions for continuity on the entire domain are established. Conditions involving the digits of the parameter $a$ and the sequence of defining functions $(\varphi_n)$, under which the function $f_a$ admits finite or continuum cardinality level sets are obtained.

For particular cases ($s=2$), we study integral and differential properties, as well as the fractal properties of the sets of values. Using the self-similarity properties of the function graph and the established connection between the functions under consideration and the inversor of digits of the $Q_2$-representation of numbers, we compute the Lebesgue integral of these functions. Furthermore, we identify a subclass of functions that are piecewise singular or singular on intervals; that is, continuous non-constant functions whose derivative is zero almost everywhere in the sense of Lebesgue measure.

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