  It is proved that the decrease of the quantum relative entropy under action
of a quantum operation is a lower semicontinuous function of a pair of its
arguments. This property implies, in particular, that the local discontinuity
jumps of the quantum relative entropy do not increase under action of quantum
operations. It implies also the lower semicontinuity of the modulus of the
joint convexity of the quantum relative entropy (as a function of ensembles of
quantum states).
  Various corollaries and applications of these results are considered.
