A topological group is locally pseudocompact if it contains a non-empty open
set with pseudocompact closure. In this note, we study connectedness and
disconnectedness properties of groups G with the property that every closed
subgroup of G is locally pseudocompact. We show that the completion of the
component G_0 of G contains every connected compact subgroup of the completion
of G. We also prove that the question of whether G/G_0 is zero-dimensional (or
equivalently, whether G_0 is dense in the component of the completion of G) can
be reduced to the case where G is a dense subgroup of a group of the form N x
R, where N is zero-dimensional and compact.