On discontinuity problem at fixed point

Abstract

We obtain some fixed point theorems under contractive conditions which admit discontinuity at the fixed point. Our results subsume all the known results of similar type, provide new answers to the question of continuity of contractive mappings at their fixed points, and also extend some recent results. Furthermore, we consider the fixed-circle problem on metric spaces. We obtain new fixed-circle results as the generalizations of some known fixed-circle theorems. Some applications of obtained results to discontinuous activation functions are also given.