c. Show that in the case of X = R, the preference relation that is represented 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 X, the upper and lower contours {y| y >= x} and {y| x >= y} are closed sets in X, and

Definition C4: For any x X, the sets {y| y >x} and {y| x >y} are open sets in X.