  We identify two key conditions that a subset $A$ of a poset $P$ may satisfy
to guarantee the transfer of continuity properties from $P$ to $A$. We then
highlight practical cases where these key conditions are fulfilled. Along the
way we are led to consider subsets of a given poset $P$ whose way-below
relation is the restriction of the way-below relation of $P$, which we call
way-below preserving subposets. As an application, we show that every
conditionally complete poset with the interpolation property contains a largest
continuous way-below preserving subposet. Most of our results are expressed in
the general setting of Z theory, where Z is a subset system.
