Remarks on the rings of functions which have a finite numb er of di scontinuities

Abstract

Let X be an arbitrary topological space. F (X) denotes the set of all real-valued functions on X and C(X)F denotes the set of all f ∈ F (X) such that f is discontinuous at most on a finite set. It is proved that if r is a positive real number, then for any f ∈ C(X)F which is not a unit of C(X)F there exists g ∈ C(X)F such that g 6= 1 and f = g f . We show that every member of C(X)F is continuous on a dense open subset of X if and only if every non-isolated point of X is nowhere dense. It is shown that C(X)F is an Artinian ring if and only if the space X is finite. We also provide examples to illustrate the results presented herein. 2010 MSC: 54C40; 13C99.