In this paper, some discontinuity results are obtained using the number MC(t, t∗) defined as MC(t, t∗) = max { d(t, t∗), ad(t,Tt) + (1 − a)d(t∗,St∗), (1 − a)d(t,Tt) + ad(t∗,St∗), b2 [d(t,St∗) + d(t∗,Tt)] } , at the common fixed point. Our results provide a new and distinct solution to an open problem “What are the contractive conditions which are strong enough to generate a fixed point but which do not force the map to be continuous at fixed point?” given by Rhoades [33]. To do this, we investigate a new discontinuity theorem at the common fixed point on a complete metric space. Also an application to threshold activation function is given