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


M. R. Ahmadi Zand, An algebraic characterization of Blumberg spaces, Quaest. Math. 33, no. 2 (2010), 223-230.

A. J. Berrick and M. E. Keating, An Introduction to Rings and Modules, Cambridge University Press, 2000.

W. Dunham, T1/2 -spaces, Kyungpook Math. J. 17, no. 2 (1977), 161-169.

R. Engelking, General Topology, Sigma Ser. Pure Math. 6, Heldermann-Verlag, Berlin, 1989.

Z. Gharabaghi, M. Ghirati. and A. Taherifar, On the rings of functions which are discontinuous on a finite set, Houston J. Math. 44, no. 2 (2018), 721-739.

L. Gillman and M. Jerison, Rings of Continuous Functions, Springer-Verlag, New York-Heidelberg, 1976.

N. Levine, Generalized closed sets in topology. Rend. Circ. Mat. Palermo. 19, no. 2 (1970), 89-96.




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.



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