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 $\succcurlyeq$ defined on $X$, without imposing continuity constraints. We show that whenever $\succcurlyeq$ has a utility representation, $f^{}_P$ is extendable if and only if it is gap-safe increasing. A class of extensions involving an arbitrary utility representation of $\succcurlyeq$ is proposed and investigated. Connections to related topological results are discussed. The condition of extendability and the form of the extension are simplified when $P$ is a Pareto set.

翻译：我们考虑将定义在任意集合$X$的子集$P$上的函数$f^{}_P$，在不施加连续性约束的条件下，严格单调地延拓到整个$X$上（相对于定义在$X$上的预序$\succcurlyeq$）。我们证明：只要$\succcurlyeq$存在效用表示，则$f^{}_P$可延拓当且仅当它是“间隙安全递增”的。本文提出并研究了一类涉及$\succcurlyeq$的任意效用表示的延拓方法，并讨论了与相关拓扑结果的联系。当$P$为帕累托集时，可延拓条件及延拓形式均可简化。