  We describe a modification of the Alicki-Fannes-Winter method which allows to
prove uniform continuity on the set of quantum states with bounded energy of
any locally almost affine function having limited growth with increasing
energy.
  Some applications in quantum information theory are considered. The
asymptotic continuity of the relative entropy of entanglement and of its
regularization under the energy constraint on one subsystem is proved.
Channel-independent continuity bounds for the quantum mutual information at the
output of a local channel and for the output Holevo quantity under the input
energy constraint are obtained.
