We study a class of functions $F$ satisfying a condition denoted by $0_x$, meaning $F(x)=0$ whenever $x$ meets a specific threshold. We show that $F$ is generated by a finite set of functions, none of which are majority or choice functions. Key theoretical results are presented, including proofs of the closure of $F$ under superposition and the inclusion $F\subseteq T_0$, where $T_0$ denotes the set of all functions that evaluate to zero whenever at least one variable is zero. Our main theorems confirm that majority functions and choice functions are not elements of $F$. We further prove that $F$ is finitely generated, that is, definable by a finite set of functions. These findings offer a novel framework for constructing finitely generated classes in multi-valued logic while excluding the commonly used but computationally intricate majority and choice functions, with practical implications in computational optimization. In conclusion, we discuss potential directions for further research, including the exploration of other classes of functions and the investigation of their properties and applications.