It has been proved here and here that a function $\mathbb{R} \to \mathbb{R}$ which has a closed and connected graph is continuous. This fact is also proved in a nice article by Burgess. I don't know how to generalize these proofs to the case of a function $\mathbb{R}^n \to \mathbb{R}$, $n \ge 2$, however. Is a function $f : \mathbb{R}^n \to \mathbb{R}$ which has a closed and connected graph necessarily continuous?
The following fact, which is stated in Munkres (Exercise 26.8), may be useful. For a topological space $X$ and compact Hausdorff space $Y$, a function $X \to Y$ is continuous if and only if its graph is a closed subset of $X \times Y$. So it would be enough to show that the function $f$ is bounded on compact sets.
