Ad hoc continuity of continuously representable interval orders

We present a continuity condition relative to an interval order on a topological space, newly defined and called strong continuity, which is analogous to the continuity of total preorders, according to which the strict lower and upper sections are open sets. Based on this kind of continuity, which is necessary for the existence of a continuous representation of the interval order on a topological space by means of two real-valued functions, we discuss the existence of continuous weak utilities for the strict part of the traces. The interest of our results is related to the consideration that the maximization of a weak utility for the strict part of any of the traces leads to maximal elements for the interval order.