## A generalization of Bernstein-Doetsch theorem.(English)Zbl 1260.26014

Summary: Let $$V$$ be an open convex subset of a nontrivial real normed space $$X$$. We give a partial generalization of the Bernstein-Doetsch theorem. We prove that if there exist a base $${\mathcal B}$$ of $$X$$ and a point $$x\in V$$ such that a midconvex function $$f: X\to{\mathbb R}$$ is locally bounded above on $$b$$-ray at $$x$$ for each $$b\in{\mathcal B}$$, then $$f$$ is convex. Moreover, we show that under the above assumption, $$f$$ is also continuous in case $$X={\mathbb R}^N$$, but not in general.

### MSC:

 26B25 Convexity of real functions of several variables, generalizations

### Keywords:

Bernstein-Doetsch theorem; midconvex function