# Continuous utility function

A rational preferences (that is, complete and transitive) are continuous. Then how can I show that there exist a continuous function u(x) that represents there preferences.

Continuity of preferences is that the upper contour sets of the bundle x and the lower contour sets of x are closed.

• You might need convexity in there somewhere. Commented Oct 31, 2017 at 20:30
• This may help:pitt.edu/~luca/ECON2100/lecture_03.pdf Commented Nov 1, 2017 at 2:49

A rational preference relation is complete and transitive, but need not be continuous. A well-known example is the lexicographic preference relation on $\mathbf{R}^2$: $(x_1,y_1) \succ (x_2,y_2)$ if and only if $x_1 > x_2$ or if ($x_1 = x_2$ and $y_1 > y_2$).

There are several theorems (the first one goes back to Debreu, 1954) proving that continuity of the preference relation is a sufficient condition for its being representable by a continuous utility function.

For a clever approach based on measure-theoretic arguments simpler than the assumption of continuity, see the paper by M. Voorneveld and J.W. Weibull (2016), "An elementary proof that well-behaved utility functions exist", Theoretical Economics Letters 6, 450--457.