Approximately convex functions on topological vector spaces

Let X be a real topological vector space, let D be a subset of X and let ® : X ! [0;1) be an even function locally bounded at zero. A function f : D ! R is called (®; t)-preconvex (where t 2 (0; 1) is ¯xed), if f(tx + (1 i t)y) · tf (x) + (1 i t)f(y) + ®(x i y) for x; y 2 D such that [x; y] ½ D: We prove the Bernstein{Doetsch type theorem for (®; t)-preconvex functions.