A remark on a utility representation theorem of Rader

Summary In this note we consider some problems involved in proving the existence of a continuous real-valued utility function representing a preference relation. We claim that there is an error in the classical Rader proof of the existence of an upper semicontinuous utility function. We also pose some open questions regarding some problems in utility theory.The work for this paper was begun while I was visiting the Department of Applied Economics at the University of Cambridge in 1992. I thank the Department for its hospitality. I should also like to thank A. F. Beardon for some extremely valuable discussions about the general nature of utility functions.