Transporting continuity properties from a poset to its subposets

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 $P$ 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.