On approximately (k,h)-convex functions

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