Some analytical properties of gamma-convex functions on the real line

This paper deals with the analytical properties of gamma-convex functions, which are defined as those functions satisfying the inequality f(x'(1))+f(x'(2))less than or equal to f(x(1))+f(x(2)), for x'(i) is an element of[x(1), x(2)]; \\x(i)-x'(1)\\ = gamma, i = 1, 2, whenever \\x(1)-x(2)\\ > gamma, for some given positive gamma. This class contains all convex functions and all periodic functions with period gamma. In general, gamma-convex functions do not have ideal properties as convex functions. For instance, there exist gamma-convex functions which are totally discontinuous or not locally bounded. But gamma-convex functions possess so-called conservation properties, meaning good properties which remain true on every bounded interval or even on the entire domain, if only they hold true on an arbitrary closed interval with length gamma. It is shown that boundedness, bounded variation, integrability, continuity, and differentiability almost everywhere are conservation properties of gamma-convex functions on the real line. However, gamma-convex functions have also infection properties, meaning bad properties which propagate to other points, once they appear somewhere (for example, discontinuity). Some equivalent properties of gamma-convexity are given. Ways for generating and representing gamma-convex functions are described.