Selection Theorems for Non-Lower Semicontinuous Mappings

Abstract

While lower semicontinuity of a mapping with closed convex values is sufficient for the existence of continuous selections, it is, of course, not necessary. For example, one can start by arbitrary continuous singlevalued map f : X→Y and then define F(x) to be a subset of Y such that f (x) ∈ F(x). Then f is a continuous selection for F, but there are no continuity type restrictions for F.