Summary: 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 \textit{\(C\)-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 à la Nachbin.