Auxiliary relations and sandwich theorems

A well-known topological theorem due to Katv etov states: Suppose $(X,tau)$ is a normal topological space, and let $f:Xto[0,1]$ be upper semicontinuous, $g:Xto[0,1]$ be lower semicontinuous, and $fleq g$. Then there is a continuous $h:Xto[0,1]$ such that $fleq hleq g$. We show a version of this theorem for many posets with auxiliary relations. In particular, if $P$ is a Scott domain and $f,g:Pto[0,1]$ are such that $fleq g$, and $f$ is lower continuous and $g$ Scott continuous, then for some $h$, $fleq hleq g$ and $h$ is both Scott and lower continuous. As a result, each Scott continuous function from $P$ to $[0,1]$, is the sup of the functions below it which are both Scott and lower continuous