With lexicographic preferences, the utility of certain goods is infinitesimal in comparison to others.
Lexicography refers to the compilation of dictionaries, and is meant to invoke the fact that a dictionary is organized alphabetically: with infinite attention to the first letter of each word, and only in the event of ties with attention to the second letter of each word, etc.
That is, there is no real-valued representation of a preference relation by a utility function, whether continuous or not.
Proof: suppose by contradiction that there exists a utility function U representing lexicographic preferences, e.g. over two goods.
Thus, lexicographic preferences can be represented by utility functions returning nonstandard real numbers.
For example, The nonstandard (infinitesimal) equilibrium prices for exchange can be determined for lexicographic order using standard equilibrium methods, except using nonstandard reals as the range of both utilities and prices.
All the theorems regarding existence of prices and equilibria extend to the case of nonstandard utilities, since the nonstandard reals form a conservative extension, meaning that any theorem which is true for reals can be extended to the nonstandard reals and remains true.