On the representability of Cr preference relations by utility functions

Summary We give a complete proof of Mas-Colell's 7] result that a continuous transitive preference relation on an open setV ^l' that isC ^r (in the sense that the indifference relation is aC ^r hypersurface inV xV) is representable by a C^r utility function without critical points.Without implicating them in remaining errors, I would like to acknowledge the helpful comments of Ket Richter and an anonymous referee.