Lower semicontinuity of the relative entropy disturbance and its corollaries

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.