The closed graph function from plane with the network of discontinuity points

Abstract

The main result of this paper states, that if a function $f:{\mathbb{R}}^2\to [0,+\infty )$ 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 $f:{\mathbb{R}}^2\to Y$ where $Y$ is a $\sigma$-locally compact metrisable space.