The Existence and Continuity of Utility Functions: A New Proof

It is well known that the existence of a countable order dense subset is necessary and sufficient for a preference order to be representable by a utility function, and that this condition is also sufficient for the utility function to be continuous with respect tot he order topology. While the modern proof of the first part of this result is based on a theorem of Cantor on ordered sets, the proof of continuity is usually based on a theorem of Debreu in real analysis. This paper seeks to eliminate this appeal to real analysis, and show that the proof of continuity requires only the order structure of the reals and does not need any metric or algebraic properties of the reals. We also show that any continuous preference ordering on a separable topological space with an at most countable number of connected components is representable by a continuous utility function thereby relaxing the usual assumption that the space be connected.