Definition:Polynomial Function/Real/Definition 2

Definition
Let $S \subset \R$ be a subset of the real numbers. Let $\R[X]$ be the polynomial ring in one variable over $\R$.

Let $\R^S$ be the ring of functions from $S$ to $\R$.

Let $\operatorname{id} \in \R^S$ 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 $\R[X] \to \R^S$ at $\operatorname{id}$.

Also see

 * Equivalence of Definitions of Real Polynomial Function