Continuity of convex functions

In this paper, we consider the set $V(K)$ of all convex real-valued functions defined on convex compacts $K\subset\mathbb R^n$ and find conditions under which all functions $f\in V(K)$ are scattered continuous. It is shown that there exist functions $f\in V(K)$ that are not Borel, and, for any ordinal $\alpha<\omega_1$, there are functions $f\in V(K)$ that exactly belong to the $\alpha$th Baire class.