  We study the inverse problem of inferring the state of a finite-level quantum
system from expected values of a fixed set of observables, by maximizing a
continuous ranking function. We have proved earlier that the maximum-entropy
inference can be a discontinuous map from the convex set of expected values to
the convex set of states because the image contains states of reduced support,
while this map restricts to a smooth parametrization of a Gibbsian family of
fully supported states. Here we prove for arbitrary ranking functions that the
inference is continuous up to boundary points. This follows from a continuity
condition in terms of the openness of the restricted linear map from states to
their expected values. The openness condition shows also that ranking functions
with a discontinuous inference are typical. Moreover it shows that the
inference is continuous in the restriction to any polytope which implies that a
discontinuity belongs to the quantum domain of non-commutative observables and
that a geodesic closure of a Gibbsian family equals the set of maximum-entropy
states. We discuss eight descriptions of the set of maximum-entropy states with
proofs of accuracy and an analysis of deviations.
