A necessary and suffi{}cient condition is presented for the existence
of a real continuous order-preserving function f on a topological
preordered space $\left(X,\tau,\preceq\right)$under a resonable continuity
assumption concerning the preorder $\preceq$ , called \textquotedblleft{}quasi
ICcontinuity\textquotedblright{}. Under the same continuity hypotheses,
a suffi{}cient condition is provided for the existence of a real continuous
order-preserving function in case that $\left(X,\tau\right)$ is a
completely regular space.
