Rings of functions which are discontinuous on a finite set with countable range

Consider the ring $C_c(X{)}_F$ of real valued functions which are discontinuous on a finite set with countable range. We discuss $({\mathcal{Z}}_c{)}_F$-filters on $X$ and $({\mathcal{Z}}_c{)}_F$-ideals of $C_c(X{)}_F$. We establish an analogous version of Gelfand-Kolmogoroff theorem in our setting. We prove some equivalent conditions when $C_c(X{)}_F$ is a Baer-ring and a regular ring. Lastly, we talk about the zero divisor graph on $C_c(X{)}_F$.