\begin{definition}[Definition:Integer-Valued Function]
Let $f: S \to T$ be a function.
Let $S_1 \subseteq S$ such that $f \left({S_1}\right) \subseteq \Z$.
Then $f$ is said to be '''integer-valued on $S_1$'''.
That is, $f$ is defined as integer-valued on $S_1$ {{iff}} the image of $S_1$ under $f$ lies entirely within the set of integers $\Z$.
An '''integer-valued function''' is a function $f: S \to \Z$ whose codomain is the set of integers $\Z$.
That is, $f$ is '''integer-valued''' {{iff}} it is '''integer-valued''' over its entire domain.
Category:Definitions/Integers
\end{definition}