On the theory of set-valued maps of bounded variation of one real variable

(Set-valued) maps of bounded variation in the sense of Jordan defined on a subset of the real line and taking values in metric or normed linear spaces are studied. A structure theorem (more general than the Jordan decomposition) is proved for such maps; an analogue of Helly's selection principle is established. A compact set-valued map into a Banach space that is a map of bounded variation (or a Lipschitz or an absolutely continuous map) is shown to have a continuous selection of bounded variation (respectively, Lipschitz or absolutely continuous selection).