Updating Utility Functions on Preordered Sets

We consider the problem of extending a function f P defined on a subset P of an arbitrary set X to X strictly monotonically with respect to a preorder ≽ defined on X , without imposing continuity constraints. We show that whenever ≽ has a utility representation, f P is extendable if and only if it is gap-safe increasing. This property means that whenever x ′ ≻ x , the infimum of f P on the upper contour of x ′ exceeds the supremum of f P on the lower contour of x , where x , x ′ ∈ X ˜ and X ˜ is X completed with two absolute ≽-extrema and, moreover, f P is weakly increasing. The completion of X makes the condition sufficient. The proposed method of extension is flexible in the sense that for any bounded utility representation u of ≽, it provides an extension of f P that coincides with u on a region of X that includes the set of P -neutral elements of X . An analysis of related topological theorems shows that the results obtained are not their consequences. The necessary and sufficient condition of extendability and the form of the extension are simplified when P is a Pareto set.