  We describe a generalized version of the result called quantum Dini lemma
that was used previously for analysis of local continuity of basic correlation
and entanglement measures. The generalization consists in considering sequences
of functions instead of a single function. It allows us to expand the scope of
possible applications of the method. We prove two general dominated convergence
theorems and the theorem about preserving local continuity under convex
mixtures.
  By using these theorems we obtain several convergence conditions for the
quantum relative entropy and for the mutual information of a quantum channel
considered as a function of a pair (channel, input state). A simple convergence
criterion for the von Neumann entropy is also obtained.
