Remark 2.9 implies that the lexicographic preference relation (example 1.114) cannot be represented by a utility function, since the lexicographic

preference ordering is not continuous. To verify this, assume, to the contrary, that u represents the lexicographic ordering ‰¿L on „œ2.
1. For every x1 ˆŠ „œ there exists a rational number r(x1) such that u(x1; 2) > r(x1) > u(x1; 1)
2. This defines an increasing function r from R to the set Q of rational numbers.
3. Obtain a contradiction.

2 x) OIX Figure 2.15 Constructive proof of the existence of a utility function