Continuous order-preserving functions for nontotal preorders on normal spaces

We discuss the continuous real representability of a not necessarily total preorder on a normal topological space in connection with a suitable continuity assumption, called C-continuity in this paper. We show that a topology $\tau$ on a set $X$ is normal if and only if the topological preordered space $(X, \precsim , \tau )$ is normally preordered for every $C$-continuous preorder $\precsim$ on $(X, \tau )$. We also prove that a C-continuous preorder $\precsim$ on a normal topological space $(X, \tau )$ is representable by means of a continuous order-preserving function $u$ if and only if $\precsim$ verifies a suitable separability condition \`a la Nachbin.