Continuity of preferences
Giskard is right lexicographic preferences would make the job being dicsontinuous, but the problem is to find relations with closed contour sets. Maybe an idea would be to start from constructing a preference relation  for which the only possible upper and lower contour sets are $X$ and $\emptyset$. In that case, you take the indiscrete topology, and all nets (or generalized sequences) converge to all the points in $X$. Anyway, I have not been able to construct such a relation yet.