A generalization of Bernstein-Doetsch Theorem

Anna Mureńko
2012 Demonstratio Mathematica  
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 = R N , but not in general. 2000 Mathematics Subject Classification: 26B25.
doi:10.1515/dema-2013-0362 fatcat:cr2vs3zqbfbqphb2brl7leonca