  Over the past years a theory of conjugate duality for set-valued functions
that map into the set of upper closed subsets of a preordered topological
vector space was developed. For scalar duality theory, continuity of convex
functions plays an important role. For set-valued maps different notions of
continuity exist. We will compare the most prevalent ones in the special case
that the image space is the set of upper closed subsets of a preordered
topological vector space and analyze which of the results can be conveyed from
the extended real-valued case. Moreover, we present a fundamental duality
formula for set-valued optimization, using the weakest of the continuity
concepts under consideration for a regularity condition.
