A note on the rings of functions which are discontinuous on some finite sets

In this paper, we study some properties of the ring $C(X)_F$ of all real valued functions which are continuous except on some finite subsets of $X$. We show that $C(X)_F$ is closed under uniform limit if and only if the set of all non-isolated points of $X$ is finite. We also initiate and investigate the zero divisor graph of the ring $C(X)_F$.