A qualitative description of graphs of discontinuous polynomial functions

Abstract

We prove that, if $f:\mathbb{R}^n\to\mathbb{R}$ satisfies Fréchet's functional equation \[ \Delta_h^{m+1}f(x)=0 \ \ \text{ for all }x=(x_1,\cdots,x_n),h=(h_1,\cdots,h_n) \in\mathbb{R}^n, \] and $f(x_1,\cdots,x_n)$ is not an ordinary algebraic polynomial in the variables $x_1,\cdots,x_n$, then $f$ is unbounded on all non-empty open set $U\subseteq \mathbb{R}^n$. Furthermore, the set $\overline{G(f)}^{\mathbb{R}^{n+1}}$ contains an unbounded open set.