  We extend Berge's Maximum Theorem to allow for incomplete preferences. We
first provide a simple version of the Maximum Theorem for convex feasible sets
and a fixed preference. Then, we show that if, in addition to the traditional
continuity assumptions, a new continuity property for the domains of
comparability holds, the limits of maximal elements along a sequence of
decision problems are maximal elements in the limit problem. While this new
continuity property for the domains of comparability is not generally necessary
for optimality to be preserved by limits, we provide conditions under which it
is necessary and sufficient.
