contour sets are topologically closed; The space X is second-countable, that is, it has a countable basis. The width of (the largest size of a set in which all elements are incomparable) is finite. The number of utility functions in the representation is at most the width of . Note that the guaranteed functions are semicontinuous, but not necessarily continuous, even if all upper and lower contour sets are closed. Evren and Ok say that "there does not seem to be a natural way of deriving a continuous finite multi-utility representation theorem, at least, not by using the methods adopted in this paper". See also Von Neumann-Morgenstern utility theorem Revealed preference theory deals with representing the demand function of an agent by a preference relation, or by a utility function. References Utility Economics theorems