  The problem is considered as to whether a monotone function defined on a
subset P of a Euclidean space can be strictly monotonically extended to the
whole space. It is proved that this is the case if and only if the function is
{\em separably increasing}. Explicit formulas are given for a class of
extensions which involves an arbitrary bounded increasing function. Similar
results are obtained for monotone functions that represent strict partial
orders on arbitrary abstract sets X. The special case where P is a Pareto
subset is considered.
