Corollary

Every convex function f defined over an open convex set X in $\mathbf{R}^n$ is continuous in X.

We can suppose that X is not empty. Then there exist, in X, $n + 1$ affinely independent points $a_i$ ($0 \leq i \leq n$) and the convex envelope of these points, S, contains the open non-empty set formed of the points $\sum_{i=0}^n \lambda_i a_i$ with $0 < \lambda_i < 1$ for all $i$ and $\sum_{i=0}^n \lambda_i = 1$. By II, p. 17, prop. 20, $f$ is bounded above in S and therefore is continuous.

In a topological vector space of infinite dimensions there exist, in general, linear non-continuous forms (II, p. 80, exerc. 25) and thus convex functions that are not continuous at any point.