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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
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