A note on monomials

Abstract

We study discontinuous solutions of the monomial equation $\frac{1}{n!}{\Delta }_h^{n}f(x)=f(h)$. In particular, we characterize the closure of their graph, ${\overline{G(f)}}^{{\mathbb{R}}^2}$, 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.

12 pages, submitted to a Journal