Problem 1. (Easy) The purpose of this problem is to make sure that you fully understand the basic concepts of utility representation and continuous preferences. a. Is the following statement correct: "if both U and V represent ~, then there is a strictly monotonic function f : R - R such that V(x) = f (U (ac) ) " ? b. Can a continuous preference relation be represented by a discontinuous utility function? c. Show that in the case of X = R, the preference relation that is rep- resented by the discontinuous utility function u(x) = [x] (the largest integer n such that x _ n) is not a continuous relation. d. Show that the two definitions of a continuous preference relation (C1 and C2) are equivalent to: Definition C3: For any x E X, the upper and lower contours fyl y x} and {yl x _ y} are closed sets in X, and Definition C4: For any x E X, the sets {yl y > x} and {yl x> y} are open sets in X