Discontinuity points of a function with a closed and connected graph

TL;DR

This paper proves that for functions from R^2 to a locally compact, second-countable space with a closed, connected, and locally connected graph, the set of discontinuity points also has a locally connected graph.

Contribution

It establishes a new property of the graph over discontinuity points for functions with specific topological graph conditions.

Findings

The graph over discontinuity points is locally connected.

The main result applies to functions with closed, connected, and locally connected graphs.

The space Y is assumed to be locally compact, second-countable, and metrizable.

Abstract

The main result of this paper states that for a function $f:\R^2\to Y$ with a closed, connected and locally connected graph, where $Y$ is a locally compact, second-countable metrisable space, the graph over discontinuity points remains locally connected.