Representing interval orders by a single real-valued function

Let > be an interval order on a topological space X, and let $\text{x > 1 y}$ if and only if there exists x′ with $\text{x?x′>y}$. Then >1 is a preference relation with transitive indifference, which, under suitable conditions, can be represented by a utility function v of the usual sort. In that case, maximisation of v (subject to contraints) leads to a most preferred point relative to >. Following both the Debreu-Rader and the Arrow-Hahn approaches, we discuss the existence and continuity of such a function v.