Discussion on Lechicki and Spakowski's counterexample

Abstract

It is well-known that intersection of continuous correspondences can lost the continuity property. Lechicki and Spakowski's theorem says that intersection of H-lsc functions remains H-lsc if the intersection is a bounded subset of a normed space and its interior is nonempty. Lechicki and Spakowski pointed to the importance of the boundedness assumption in the case of infinite dimensional range giving a counterexample. Even though the counterexample works properly and is one of the most cited patterns of discontinuity, it has no detailed discussion in the literature of economics and optimization theory. What is more, some misleading interpretation of this very important counterexample can be observed. Our technical note clarifies the exact role of Lechicki and Spakowski's counterexample, computing each of the important properties of the correspondences rigorously