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.
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