Let be a complete, reflexive and transitive preference relation on X. We showed in class that if X is finite, then there is a utility function u that represents . In this question we will see what happens when X is (uncountably) infinite. Suppose X is R? and is defined by (11,D) (91.ya) iff I1 > y or (11 = y1 and ra = 42). These preferences are called lexicographic.

i. Show that defines a complete, reflexive and transitive preference relation on X

ii. Show that there is no continuous utility function : x + R that rep resents. Therefore, the theorem proved in class cannot be extended to the case of infinite X.

iii. If in addition to completeness, reflexivity and transitivity of the preference relation , it also satisfies continuity and strict monotonicity, then can be represented by a utility function (you don't have to prove this). Which of these properties (continuity and strict mono tonicity) are violated by lexicographic preferences?