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A method of proving local continuity of concave functions on convex setpossessing the $\mu$-compactness property is presented. This method is based ona special approximation of these functions.  The class of $\mu$-compact sets can be considered as a natural extension ofthe class of compact metrizable subsets of locally convex spaces, to whichparticular results well known for compact sets can be generalized.  Applications of the obtained continuity conditions to analysis of differententropic characteristics of quantum systems and channels are considered.

Continuity condition for concave functions on convex $\mu$-compact sets and its applications in quantum physics