  We prove several facts about the Yamabe constant of Riemannian metrics on
general noncompact manifolds and about S. Kim's closely related "Yamabe
constant at infinity". In particular we show that the Yamabe constant depends
continuously on the Riemannian metric with respect to the fine C^2-topology,
and that the Yamabe constant at infinity is even locally constant with respect
to this topology. We also discuss to which extent the Yamabe constant is
continuous with respect to coarser topologies on the space of Riemannian
metrics.
