Please state any theorems you are using.

Question 1: For three compact metric spaces (X ,d), (Y , e) and (Z, f) suppose u :X x Y ) Z is a continuous function and (it : X >+ Y and I]! : X >> Z are two lower and upper hemi continuous correspondences. Let the correspondence 1" :X >> Y be dened by ['(x) = {y 6 (MI) | \"(36.30 6 WI) }- 1. Illustrate the situation outlined. 2. Is 1" lower-continuous for all (X , Y ,2, u, a, 1;!) satisfying the assumptions? 3. Is 1" upper hemi-continuous for all (X ,Y,Z, u, p, q!) satisfying the assumptions? 4. Suppose Z = [1,1] and Mr) = {0} for all x E X. lustrate that 1" does not need to be a singleton for all (X ,Y ,Z, u, q), 1}!) satisfying the assumptions and x E X. 5. How could the setting and upper hemi-continuity of 1" be applied to economics