A qualitative description of graphs of discontinuous polynomial functions

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.