\begin{definition}[Definition:Polynomial Function/Real/Definition 1]
Let $S \subset \R$ be a subset of the real numbers.
A '''real polynomial function''' on $S$ is a function $f: S \to \R$ for which there exist:
:a natural number $n\in \N$
:real numbers $a_0, \ldots, a_n \in \R$
such that for all $x \in S$:
:$\map f x = \ds \sum_{k \mathop = 0}^n a_k x^k$
where $\sum$ denotes indexed summation.
\end{definition}