On Approximately h-Convex Functions

A real valued function $f\colon D\to \R$ defined on an open convex subset $D$ of a normed space $X$ is called rationally $(h,d)$-convex if it satisfies $$f\left(tx + (1-t)y \right) \leq h(t) f(x) + h(1-t) f(y) + d(x,y)$$ for all $x,y\in D$ and $t\in \Q \cap [0,1]$, where $d\colon X \times X \to \R$ and $h:[0,1] \to \R$ are given functions. Our main result is of Bernstein-Doetsch type. Namely, we prove that if $f$ is locally bounded from above at a point of $D$ and rationally $(h,d)$-convex then it is continuous and $(h,d)$-convex.