Continuous Representations of Preferences by Means of Two Continuous Functions

Let ≾ be a reflexive binary relation on a topological space (X, τ ). A pair (u,v) of continuous real-valued functions on (X, τ ) is said to be a continuous representation of ≾ if, for all x,y ∈ X, [(x ≾ y ⇔ u(x) ≤ v(y))]. In this paper we provide a characterization of the existence of a continuous representation of this kind in the general case when neither the functions u and v nor the topological space (X,τ ) are required to satisfy any particular assumptions. Such characterization is based on a suitable continuity assumption of the binary relation ≾, called weak continuity. In this way, we generalize all the previous results on the continuous representability of interval orders, and also of total preorders, as particular cases.