<p>In 1971, Reich (Some remarks concerning contraction mappings. Can Math Bull 14:121–124, 1971) unified the Banach contraction principle and the Kannan contraction theorem by introducing a new class of contractions, now known as Reich contractions, which ensure the existence and uniqueness of a fixed point in a metric space. His condition permits discontinuity in the domain of definition but requires the self-mapping to be continuous at the fixed point. In this paper, we extend Reich’s fixed point theorem to include both contractive and non-expansive mappings in metric spaces. This extension admits the possibility of multiple fixed points, even when a fixed point is itself a point of discontinuity. The resulting structured fixed-point sets and domains exhibit rich algebraic, geometric, and dynamical structures. Our theorem thus provides a broad generalization of several well-known results on contractive mappings. As a byproduct, we present a new solution–distinct from the known one–to the problem posed by Rhoades (Contractive definitions and continuity. Contemp Math 72:233–245, 1988) concerning the existence of contractive mappings that admit a fixed point which is also a point of discontinuity.</p>