A generalization of Bernstein Doetsch Theorem

Let V be an open convex subset of a nontrivial real normed space X. In the paper we give a partial generalization of Bernstein-Doetsch Theorem. We prove that if there exist a base B of X and a point x ∈ V such that a midconvex function f : X → R is locally bounded above on b-ray at x for each b ∈ B, then f is convex. Moreover, we show that under the above assumption, f is also continuous in case X = RN , but not in general. Let X be a real normed space and V be a convex subset of X. A function f : V → R is called convex if f(tx + (1− t)y) ≤ tf(x) + (1− t)f(y) for x, y ∈ V, t ∈ [0, 1]. If the above inequality holds for t = 1 2 , then f is said to be midconvex (or Jensen convex). F. Bernstein and G. Doetsch [1] proved that every midconvex function f : (a, b) → R locally bounded above at a point is continuous (clearly a continuous midconvex function must be convex). The above statement has many generalizations (see e. g. [2]–[10]). Nowadays by Bernstein-Doetsch Theorem we usually mean the following one. Bernstein-Doetsch Theorem Let V be an open convex subset of X and let f : V → R be midconvex. If f is locally bounded above at a point, then f is continuous and convex. In the paper we give a generalization of Bernstein-Doetsch Theorem. To formulate this generalization, we introduce the following definition. Mathematics Subject Classification (2000): 26B25