Some results concerning the generic continuity of set-valued mappings

A theorem proved by Fort in 1951 states that an upper semi-continuous mapping from a Baire space XX into non-empty compact subsets of a metric space is both lower and upper semi-continuous at the points of a dense residual set of XX. In this paper, we present several results concerning the generic continuity of set-valued mappings. First we demonstrate that generic continuity, in the sense of Baire category, cannot be extended to almost everywhere continuity (with respect to some measure). Then we show that Baireness of the domain space is a necessary condition for Fort’s theorem to hold. Finally, we show that Fort’s theorem holds without any compactness assumption on the images of the set-valued mapping.