A note on monomials

TL;DR

This paper investigates discontinuous solutions to a specific monomial functional equation, characterizes their graph closure, and provides new proofs for classical theorems related to polynomials and additive functions.

Contribution

It offers a novel analysis of the structure of solutions to the monomial equation and introduces new proofs for established theorems using these properties.

Findings

Characterization of the closure of the graph of solutions.

New proof of the Darboux type theorem for polynomials.

New proof of Hamel's theorem for additive functions.

Abstract

We study discontinuous solutions of the monomial equation $\frac{1}{n!}\Delta_h^nf(x)=f(h)$. In particular, we characterize the closure of their graph, $\bar{G(f)}^{\mathbb{R}^2}$, and we use the properties of these functions to present a new proof of the Darboux type theorem for polynomials and of Hamel's theorem for additive functions.