  It is well known that the von Neumann entropy is continuous on a subset of
quantum states with bounded energy provided the Hamiltonian $H$ of the system
satisfies the condition $\Tr\exp(-cH)<+\infty$ for any $c>0$. In this note we
consider the following conjecture: every closed convex subset of quantum
states, on which the von Neumann entropy is continuous, consists of states with
bounded energy with respect to a particular Hamiltonian $H$ satisfying the
above condition.
  It is shown that the classical analog of this conjecture is valid (i.e. it is
valid for the Shannon entropy). It is also shown that this conjecture holds for
some types of subsets consisting of non-commuting states, but its validity for
all subsets of quantum states remains an open question.
