A qualitative description of graphs of discontinuous polynomial functions

TL;DR

This paper investigates the properties of solutions to Fréchet's functional equation, showing that non-polynomial solutions are unbounded on open sets and their graphs' closures contain unbounded open sets.

Contribution

It provides a qualitative description of the graphs of discontinuous polynomial functions satisfying Fréchet's equation, highlighting their unboundedness and topological properties.

Findings

Non-polynomial solutions are unbounded on all open sets.

The closure of the graph contains an unbounded open set.

Solutions satisfying Fréchet's equation are either polynomial or exhibit unbounded behavior.

Abstract

We prove that, if f:R^n\to R satisfies Fr\'echet's functional equation and f(x_1,...,x_n) is not an ordinary algebraic polynomial in the variables x_1,...,x_n, then f is unbounded on all non-empty open set U of R^n. Furthermore, the closure of its graph contains an unbounded open set.