A new solution to the discontinuity problem on metric spaces

Abstract

We study on the Rhoades’ question concerning the discontinuity problem at fixed point for a self-mappingT of a metric space. We obtain a new solution to this question. Our result generalizes some recent theorems existing inthe literature and implies the uniqueness of the fixed point. However, there are also cases where the fixed point set of aself-mapping contains more than 1 element. Therefore, by a geometric point of view, we extend the Rhoades’ questionto the case where the fixed point set is a circle. We also give a solution to this extended version.