  The main result of this paper states, that if a function $f:\R^2\to [0,
+\infty)$ has a closed graph and the set of discontinuity points is a network
(as defined by Kuratowski in Topology II, 61.IV), then the graph of $f$ is
disconnected. It is also proven that this result can be easily generalised to a
function $f:\R^2\to Y$ where $Y$ is a $\sigma$-locally compact metrisable
space.
