Agarwal et al [1] have proved some interesting local and global fixed point theorems for Meir-Keeler [7] type and Caristi [2] type maps. We obtain analogues of the main results of Agarwal et al [1] under weaker conditions so as to include continuous as well as discontinuous maps. Our results provide new answers to Rhoades' problem ([15], p. 242) on existence of contractive definitions which admit discontinuity at the fixed point. Several examples are given to illustrate our results.