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

TL;DR

This paper explores the properties of a ring of real-valued functions that are discontinuous only on finite sets, focusing on its closure under uniform limits and the structure of its zero divisor graph.

Contribution

It characterizes when the ring is closed under uniform limits and introduces the zero divisor graph for this class of functions.

Findings

The ring is closed under uniform limits iff the set of non-isolated points of X is finite.

Initiates the study of the zero divisor graph for C(X)_F.

Provides new insights into the algebraic structure of functions with finite discontinuities.

Abstract

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 $C(X)_F$.