Let X be a paracompact space, Y a normed vector space and

${2}^{Y}$ the collection of all non-empty subsets of Y. A function f:

$X\to {2}^{Y}$ 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

$Y={C}_{0}\left(T\right)$ and

$Y={L}_{1}$, the authors give intrinsic characterizations of the one-dimensional subspaces whose metric projections admit continuous selections.