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.

## Preferences relation from utility function

Voorneveld and J.W. Weibull (2016), "An elementary proof that well-behaved utility functions exist", *Theoretical Economics Letters***6**, 450--457.

answered Nov 4 '17 at 9:38

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