Extending Utility Functions on Arbitrary Sets

We consider the problem of whether a function $f^{}_P$ defined on a subset $P$ of an arbitrary set $X$ can be extended to $X$ monotonically with respect to a preorder $\succcurlyeq$ defined on $X$. We prove that whenever $\succcurlyeq$ has a utility representation, such an extension exists if and only if $f^{}_P$ is gap-safe increasing. An explicit construction for a monotone extension of this kind involving an arbitrary utility representation of $\succcurlyeq$ is presented. The special case where $P$ is a Pareto subset of $X$ is considered. The problem under study does not include continuity constraints.