Consider the following optimization set up and notation

Now consider the following problem:

Let , and for all . Define

Graph and show it is upper hemi continuous. Show where it fails to be lower hemi continuous.

A canonical maximization problem consists of: . a parameter set a choice set X an objective function f :X R a constraint correspondence 1:0 3X a solution correspondence T*0) := arg max f(x,0) xer(O) the maximized value of the objective function f*(0) = max f(x,0) xer(0) X = = f(0,0) = max{2 - (x - 1)2,0 +1-(x - 2)2} A canonical maximization problem consists of: . a parameter set a choice set X an objective function f :X R a constraint correspondence 1:0 3X a solution correspondence T*0) := arg max f(x,0) xer(O) the maximized value of the objective function f*(0) = max f(x,0) xer(0) X = = f(0,0) = max{2 - (x - 1)2,0 +1-(x - 2)2}