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

TL;DR

This paper proves that functions from the plane to the non-negative reals with a closed graph and a network of discontinuity points have disconnected graphs, extending the result to certain metric spaces.

Contribution

It establishes a new connection between the structure of discontinuity sets and the connectivity of the graph for functions from the plane.

Findings

Functions with a network of discontinuities have disconnected graphs.

The result generalizes to functions into σ-locally compact metrisable spaces.

Abstract

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.