An example showing that A-lower semi-continuity is essential for minimax continuity theorems

Recently the authors have established continuity properties of minimax values and solution sets for a function of two variables depending on a parameter. Some of these properties hold under the assumption that the multifunction, defining the domains of the second variable, is A-lower semi-continuous. This property is stronger than lower semi-continuity, but in several important cases these two properties coincide. This note provides an example demonstrating that in general the A-lower semi-continuity assumption cannot be relaxed to lower semi-continuity.