Let X be a paracompact space, Y a normed vector space and
the collection of all non-empty subsets of Y. A function f:
is called a set-valued mapping. The relationships between lower semicontinuity, almost lower semicontinuity and the existence of various kinds of continuous selections for such a mapping are studied. For the cases
, the authors give intrinsic characterizations of the one-dimensional subspaces whose metric projections admit continuous selections.