\begin{definition}[Definition:Real-Valued Function]
Let $f: S \to T$ be a function.
Let $S_1 \subseteq S$ such that $\map f {S_1} \subseteq \R$.
Then $f$ is said to be '''real-valued on $S_1$'''.
That is, $f$ is defined as '''real-valued on $S_1$''' {{iff}} the image of $S_1$ under $f$ lies entirely within the set of real numbers $\R$.
A '''real-valued function''' is a function $f: S \to \R$ whose codomain is the set of real numbers $\R$.
That is, $f$ is '''real-valued''' {{iff}} it is '''real-valued''' over its entire domain.
\end{definition}