Question

Suppose that a person’s preferences are such that there is a (small) ε > 0 such that x ∼ y if ∥x − y∥ < ε, but if the bundle x has more of all goods than does the bundle y, and if ∥x−y∥ ≥ ε, then x ≻ y. Show that there is no utility function that represents such preferences.

Answer 1

I think that this issue is related to lexicographic advantages, since it does not have a useful function.