# Definition:Real Function/Domain

## Definition

Let $S \subseteq \R$.

Let $f: S \to \R$ be a real function.

The domain of $f$ is the set $S$.

It is frequently the case that $S$ is not explicitly specified. If this is so, then it is understood that the domain is to consist of all the values in $\R$ for which the function is defined.

This often needs to be determined as a separate exercise in itself, by investigating the nature of the function in question.

## Also known as

Some treatments of the subject consider domains of limited generality: for example, closed intervals, and consequently specify such an interval $\set {x \in \R: a \le x \le b}$ as $x$-space.