In the present paper we prove a fixed point theorem for a one parameter family of contractive self-mappings, of a complete metric space or a complete b-metric space, each member of which has a unique fixed point that is also the unique common fixed point of the family; the mappings may be continuous or discontinuous at the fixed point. We also prove that under the assumption of a weaker form of continuity the fixed point property for mappings satisfying the contractive conditions employed by us implies completeness of the underlying space. The characterization of completeness obtained by us not only contains Subrahmanyam's theorem on characterization of completeness as a particular case but also extends it to b-metric spaces. Results on contractive mappings with discontinuity at the fixed point have found applications in neural networks with discontinuous activation function (e.g. Ozgur and Tas [19, 20]).