\begin{definition}[Definition:Real Function/Definition 1]
Let $S \subseteq \R$ be a subset of the set of real numbers $\R$.
Suppose that, for each value of the independent variable $x$ of $S$, there exists a corresponding value of the dependent variable $y$.
Then the dependent variable $y$ is a '''(real) function''' the independent variable $x$.
\end{definition}