New results on discontinuity at fixed point

We obtain a Meir-Keeler type fixed-point theorem which gives a new solution to the Rhoades' problem on the existence of contractive mappings that admit discontinuity at the fixed point. Meir-Keeler type solutions of the Rhoades' problem have not been reported in literature before this. Presenting a new approach, we give another solution to this problem using the set of simulation functions. To emphasize the importance of our theoretical results, we obtain two applications of the main results with some illustrative examples.