Definition:Integer-Valued Function

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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 $S_1$ if and only if 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 if and only if it is integer-valued over its entire domain.