In this paper we continue the study of the general J-convex functions, which are introduced in our former paper (Tasković, Math. Japonica, 37 (1992), 367-372). We prove that if $D \subset \mathbb{R}^n$ a convex and open set, and if $f: D \to \mathbb{R}$ is a general J-inner function with the property of local oscillation in $D$, then it is continuous in $D$. Since every J-convex function (also an additive function) is general J-inner function, we obtain as a particular case of the preceding statement the result of F. Bernstein and G. Doetsch.