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




C(X)F, Z-ultrafilter, completely separated, C(X)F -embedded, Z-filter, over-rings of C(X), Artinian ring


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 ≠ 1 and f = gr 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.


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Author Biographies

Mohammad Reza Ahmadi Zand, Yazd University

Department of Mathematics

Zahra Khosravi, Yazd University

Department of Mathematics


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How to Cite

M. R. Ahmadi Zand and Z. Khosravi, “Remarks on the rings of functions which have a finite numb er of di scontinuities”, Appl. Gen. Topol., vol. 22, no. 1, pp. 139–147, Apr. 2021.