  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.
