General methods of quantitative and qualitative continuity analysis of
    characteristics of composite quantum systems are described. Several
    modifications of the Alicki-Fannes-Winter method are considered, which make it
    applicable to a wide class of characteristics in both finite-dimensional and
    infinite-dimensional cases. A new approximation method for obtaining local
    continuity conditions for various characteristics of quantum systems is
    proposed and described in detail. This method allows us to prove several
    general results (Simon-type dominated convergence theorem, the theorem about
    preserving continuity under convex mixtures, etc.).

    Uniform continuity bounds and local continuity conditions for basic
    characteristics of composite quantum systems are presented. Along with the
    results obtained earlier by different authors, a number of new results proved
    by the proposed methods are described.

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