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We consider a quasi-classical version of the Alicki-Fannes-Winter techniquewidely used for quantitative continuity analysis of characteristics of quantumsystems and channels. This version allows us to obtain continuity bounds underconstraints of different types for quantum states belonging to subsets of aspecial form that can be called "quasi-classical".  Several applications of the proposed method are described. Among others, weobtain the universal continuity bound for the von Neumann entropy under theenergy-type constraint which in the case of one-mode quantum oscillator isclose to the specialized optimal continuity bound presented recently by Becker,Datta and Jabbour.  We obtain semi-continuity bounds for the quantum conditional entropy ofquantum-classical states and for the entanglement of formation in bipartitequantum systems with the rank/energy constraint imposed only on one state.Semi-continuity bounds for entropic characteristics of classical randomvariables and classical states of a multi-mode quantum oscillator are alsoobtained.

Close-to-optimal continuity bound for the von Neumann entropy and other quasi-classical applications of the Alicki-Fannes-Winter technique