### On the continuity of Q-convex functions and additive functions.

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The notion of strongly n-convex functions with a control symmetric ()n$-linearfunction(\varphi $ in the class of functions acting from one real linear space to another one are introduced. Some connections between such functions and $n$-convex functions are also given.

We generalize the well known separation theorems for subadditive and superadditive functionals to some classes of not necessarily Abelian semigroups. We also consider the problem of supporting subadditive functionals by additive ones in the not necessarily commutative case. Our results are motivated by similar extensions of the Hyers stability theorem for the Cauchy functional equation. In this context the so-called weakly commutative and amenable semigroups appear naturally. The relations between...

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