ABSTRACT. Some new continuity concepts for metric projections are introduced which are simpler and more general than the usual upper semicontinuity and lower semicontinuity. These concepts are strong enough to generalize a number of known results, yet weak enough so that now the converses of many of these generalizations are also valid. In particular, in a large class of normed linear spaces, suns and Chebychev sets can be characterized by a certain continuity property of their metric projections.