The authors define and study a common generalization of UC spaces and boundedly compact spaces, the boundedly UC spaces. A UC space is a metric space on which every continuous map is uniformly continuous and a boundedly compact space is one in which closed and bounded sets are compact. Boundedly UC spaces are metric spaces for which continuous maps are uniformly continuous on bounded sets.
A host of equivalences is provided as well as the result that a metrizable space admits a boundedly UC metric if{}f its set of nonisolated points is locally compact and separable. The final result equates, for boundedly UC spaces, the topology of uniform convergence on bounded sets and a certain hyperspace topology on the set of graphs of continuous maps considered by H. Attouch and R. Wets [Trans. Am. Math. Soc. 328, No. 2, 695-729 (1991; Zbl 0753.49007)].