Continuity of set-valued maps revisited in the light of tame geometry

TL;DR

This paper revisits the continuity properties of set-valued maps within tame geometry, establishing that semialgebraic set-valued maps are almost everywhere continuous and extending results to o-minimal structures.

Contribution

It proves that semialgebraic set-valued maps are almost everywhere continuous, extending to o-minimal structures, and provides Sard type results for local minima.

Findings

Semialgebraic set-valued maps are almost everywhere continuous.

Results extend to o-minimal (tame) set-valued maps.

Includes Sard type results for local minima.

Abstract

Continuity of set-valued maps is hereby revisited: after recalling some basic concepts of variational analysis and a short description of the State-of-the-Art, we obtain as by-product two Sard type results concerning local minima of scalar and vector valued functions. Our main result though, is inscribed in the framework of tame geometry, stating that a closed-valued semialgebraic set-valued map is almost everywhere continuous (in both topological and measure-theoretic sense). The result, depending on stratification techniques, holds true in a more general setting of o-minimal (or tame) set-valued maps. Some applications are briefly discussed at the end.