\begin{definition}[Definition:Polynomial Function/Real/Definition 2]
Let $S \subset \R$ be a subset of the real numbers.
Let $\R \sqbrk X$ be the polynomial ring in one variable over $\R$.
Let $\R^S$ be the ring of mappings from $S$ to $\R$.
Let $\iota \in \R^S$ denote the inclusion $S \hookrightarrow \R$.
A '''real polynomial function''' on $S$ is a function $f: S \to \R$ which is in the image of the evaluation homomorphism $\R \sqbrk X \to \R^S$ at $\iota$.
\end{definition}