# Definition:Real Function/Domain

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## 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 $\left\{ {x \in \R: a \le x \le b}\right\}$ as **$x$-space**.

## Sources

- 1964: William K. Smith:
*Limits and Continuity*... (previous) ... (next): $\S 2.2$: Functions - 1968: A.N. Kolmogorov and S.V. Fomin:
*Introductory Real Analysis*... (previous) ... (next): $\S 1.3$: Functions and mappings. Images and preimages - 1968: Peter D. Robinson:
*Fourier and Laplace Transforms*... (next): $\S 1.1$. The Idea of an Integral Transform