Remarks on the rings of functions which have a finite numb er of di scontinuities
Authors: Ahmadi Zand Mohammad Reza , Khosravi Zahra 
Published: April 2021

<p>Let X be an arbitrary topological space. F(X) denotes the set of all real-valued functions on X and C(X)<sub>F</sub> 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)<sub>F</sub> which is not a unit of C(X)<sub>F</sub> there exists g ∈ C(X)<sub>F</sub> such that g ≠ 1 and f = g<sup>r</sup> f. We show that every member of C(X)<sub>F</sub> 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)<sub>F</sub> is an Artinian ring if and only if the space X is finite. We also provide examples to illustrate the results presented herein.</p><p><span style="font-family: CMR10; font-size: 10pt; color: #000000; font-style: normal; font-variant: normal;"><span style="font-family: CMMI10; font-size: 10pt; color: #000000; font-style: normal; font-variant: normal;"><span style="font-family: CMR10; font-size: 10pt; color: #000000; font-style: normal; font-variant: normal;"><span style="font-family: CMMI10; font-size: 10pt; color: #000000; font-style: normal; font-variant: normal;"><span style="font-family: CMR10; font-size: 10pt; color: #000000; font-style: normal; font-variant: normal;"><span style="font-family: CMMI10; font-size: 10pt; color: #000000; font-style: normal; font-variant: normal;"><span style="font-family: CMR10; font-size: 10pt; color: #000000; font-style: normal; font-variant: normal;"><span style="font-family: CMR10; font-size: 10pt; color: #000000; font-style: normal; font-variant: normal;"><span style="font-family: CMMI10; font-size: 10pt; color: #000000; font-style: normal; font-variant: normal;"><span style="font-family: CMR10; font-size: 10pt; color: #000000; font-style: normal; font-variant: normal;"><span style="font-family: CMMI10; font-size: 10pt; color: #000000; font-style: normal; font-variant: normal;"><span style="font-family: CMR10; font-size: 10pt; color: #000000; font-style: normal; font-variant: normal;"><span style="font-family: CMMI10; font-size: 10pt; color: #000000; font-style: normal; font-variant: normal;"><span style="font-family: CMR10; font-size: 10pt; color: #000000; font-style: normal; font-variant: normal;"><span style="font-family: CMMI8; font-size: 7pt; color: #000000; font-style: normal; font-variant: normal;"><span style="font-family: CMR10; font-size: 10pt; color: #000000; font-style: normal; font-variant: normal;"><span style="font-family: CMMI10; font-size: 10pt; color: #000000; font-style: normal; font-variant: normal;"><span style="font-family: CMSY10; font-size: 10pt; color: #000000; font-style: normal; font-variant: normal;"><span style="font-family: CMMI10; font-size: 10pt; color: #000000; font-style: normal; font-variant: normal;"><span style="font-family: CMR10; font-size: 10pt; color: #000000; font-style: normal; font-variant: normal;"><span style="font-family: CMMI10; font-size: 10pt; color: #000000; font-style: normal; font-variant: normal;"><span style="font-family: CMR10; font-size: 10pt; color: #000000; font-style: normal; font-variant: normal;"><span style="font-family: CMMI10; font-size: 10pt; color: #000000; font-style: normal; font-variant: normal;"><span style="font-family: CMR10; font-size: 10pt; color: #000000; font-style: normal; font-variant: normal;"><span style="font-family: CMR10; font-size: 10pt; color: #000000; font-style: normal; font-variant: normal;"><span style="font-family: CMMI10; font-size: 10pt; color: #000000; font-style: normal; font-variant: normal;"><span style="font-family: CMR10; font-size: 10pt; color: #000000; font-style: normal; font-variant: normal;"><span style="font-family: CMMI10; font-size: 10pt; color: #000000; font-style: normal; font-variant: normal;"><span style="font-family: CMSY10; font-size: 10pt; color: #000000; font-style: normal; font-variant: normal;"><span style="font-family: CMMI10; font-size: 10pt; color: #000000; font-style: normal; font-variant: normal;"><span style="font-family: CMR10; font-size: 10pt; color: #000000; font-style: normal; font-variant: normal;"><span style="font-family: CMMI10; font-size: 10pt; color: #000000; font-style: normal; font-variant: normal;"><span style="font-family: CMR10; font-size: 10pt; color: #000000; font-style: normal; font-variant: normal;"><span style="font-family: CMMI8; font-size: 7pt; color: #000000; font-style: normal; font-variant: normal;"><span style="font-family: CMR10; font-size: 10pt; color: #000000; font-style: normal; font-variant: normal;"><span style="font-family: CMMI10; font-size: 10pt; color: #000000; font-style: normal; font-variant: normal;"><span style="font-family: CMR10; font-size: 10pt; color: #000000; font-style: normal; font-variant: normal;"><span style="font-family: CMMI10; font-size: 10pt; color: #000000; font-style: normal; font-variant: normal;"><span style="font-family: CMR10; font-size: 10pt; color: #000000; font-style: normal; font-variant: normal;"><span style="font-family: CMMI8; font-size: 7pt; color: #000000; font-style: normal; font-variant: normal;"><span style="font-family: CMR10; font-size: 10pt; color: #000000; font-style: normal; font-variant: normal;"><span style="font-family: CMMI10; font-size: 10pt; color: #000000; font-style: normal; font-variant: normal;"><span style="font-family: CMSY10; font-size: 10pt; color: #000000; font-style: normal; font-variant: normal;"><span style="font-family: CMMI10; font-size: 10pt; color: #000000; font-style: normal; font-variant: normal;"><span style="font-family: CMR10; font-size: 10pt; color: #000000; font-style: normal; font-variant: normal;"><span style="font-family: CMMI10; font-size: 10pt; color: #000000; font-style: normal; font-variant: normal;"><span style="font-family: CMR10; font-size: 10pt; color: #000000; font-style: normal; font-variant: normal;"><span style="font-family: CMMI8; font-size: 7pt; color: #000000; font-style: normal; font-variant: normal;"><span style="font-family: CMR10; font-size: 10pt; color: #000000; font-style: normal; font-variant: normal;"><span style="font-family: CMMI10; font-size: 10pt; color: #000000; font-style: normal; font-variant: normal;"><span style="font-family: CMSY10; font-size: 10pt; color: #000000; font-style: normal; font-variant: normal;"><span style="font-family: CMR10; font-size: 10pt; color: #000000; font-style: normal; font-variant: normal;"><span style="font-family: CMMI10; font-size: 10pt; color: #000000; font-style: normal; font-variant: normal;"><span style="font-family: CMR10; font-size: 10pt; color: #000000; font-style: normal; font-variant: normal;"><span style="font-family: CMMI10; font-size: 10pt; color: #000000; font-style: normal; font-variant: normal;"><span style="font-family: CMMI8; font-size: 7pt; color: #000000; font-style: normal; font-variant: normal;"><span style="font-family: CMMI10; font-size: 10pt; color: #000000; font-style: normal; font-variant: normal;"><span style="font-family: CMR10; font-size: 10pt; color: #000000; font-style: normal; font-variant: normal;"><span style="font-family: CMMI10; font-size: 10pt; color: #000000; font-style: normal; font-variant: normal;"><span style="font-family: CMR10; font-size: 10pt; color: #000000; font-style: normal; font-variant: normal;"><span style="font-family: CMMI10; font-size: 10pt; color: #000000; font-style: normal; font-variant: normal;"><span style="font-family: CMR10; font-size: 10pt; color: #000000; font-style: normal; font-variant: normal;"><span style="font-family: CMMI8; font-size: 7pt; color: #000000; font-style: normal; font-variant: normal;"><span style="font-family: CMR10; font-size: 10pt; color: #000000; font-style: normal; font-variant: normal;"><span style="font-family: CMR10; font-size: 10pt; color: #000000; font-style: normal; font-variant: normal;"><span style="font-family: CMMI10; font-size: 10pt; color: #000000; font-style: normal; font-variant: normal;"><span style="font-family: CMR10; font-size: 10pt; color: #000000; font-style: normal; font-variant: normal;"><br style="font-style: normal; font-variant: normal; font-weight: normal; letter-spacing: normal; line-height: normal; orphans: 2; text-align: -webkit-auto; text-indent: 0px; text-transform: none; white-space: normal; widows: 2; word-spacing: 0px; -webkit-text-size-adjust: auto; -webkit-text-stroke-width: 0px;" /></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></p>