Some Analytical Properties of γ-Convex Functions in Normed Linear Spaces.

For a fixed positive number γ, a real-valued function f defined on a convex subset D of a normed space X is said to be γ-convex if it satisfies the inequality f(x′₀) + f (x′₁) ⩽ f(x₀) + f(x₁), for x′_i ∊ [x₀, x₁], for x′_i ∊[x₀, x₁], ∥x′_i - x_i∥ = γ, i = 0, 1, whenever x₀, x₁ ∊ D and ∥x₀ - x₁∥ ➮ γ. This paper presents some results on the boundedness and continuity of γ-convex functions. For instance, (a) if there is some x_* ∊ D such that f is bounded below on D ∩ B (x_*, γ), then so it is on each bounded subset of D; (b) if f is bounded on some closed ball B(x_*, γ/2) ⊂ D and D′ is a closed bounded subset of D, then f is bounded on D′ if it is bounded above on the boundary of D′; (c) if dim X > 1 and the interior of D contains a closed ball of radius γ, then f is either locally bounded or nowhere locally bounded in the interior of D; (d) if D contains some open ball B(x_*, γ/2) in which f has at most countably many discontinuities, then the set of all points at which f is continuous is dense in D. [ABSTRACT FROM AUTHOR]