Some properties of upper semicontinuous multivalued mappings (Q921562)

Two interesting examples are presented throwing some light on the lack of regularity which upper semicontinuous (u.s.c.) set-valued mappings may possess. Both constructed u.s.c. multifunctions are defined on [0,1] with convex, compact values in \({\mathbb{R}}^ n\). In the first situation discussed the sets of points of discontinuity for the multifunction itself and every selection are of full measure. The second construction shows a set-valued mapping with a given \(F_{\sigma}\)-type set E of points of discontinuity and such that all selections are discontinuous on E as well. Both examples are complemented with some additional remarks. The above-mentioned results have analogues in the qualitative theory of optimal controls for linear control systems and for linear differential games. Namely, there are situations when the set of points of discontinuity for optimal controls or optimal strategies may be of full measure.