A dichotomy property for the graphs of monomials

TL;DR

This paper establishes a dichotomy for the graphs of discontinuous monomials, showing they are either connected or totally disconnected, and characterizes those with connected graphs.

Contribution

It proves a dichotomy property for the graphs of discontinuous monomials and characterizes connected graphs among them, extending to additive functions in multiple dimensions.

Findings

Discontinuous monomials have either connected or totally disconnected graphs.

Connected graphs of monomials satisfy a specific big graph property.

The paper analyzes connectedness properties of additive functions' graphs.

Abstract

We prove that the graph of a discontinuous $n$-monomial function $f:\mathbb{R}\to\mathbb{R}$ is either connected or totally disconnected. Furthermore, the discontinuous monomial functions with connected graph are characterized as those satisfying a certain big graph property. Finally, the connectedness properties of the graphs of additive functions $f:\mathbb{R}^d\to\mathbb{R}$ are studied.