# Definition:Real-Valued Function

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## Definition

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

## Also known as

Some sources refer to this as a **numerical function defined in $S_1$**.

## Also see

- Definition:Real Function, in which the domain and codomain are both subsets of $\R$.

## Sources

- 1965: Claude Berge and A. Ghouila-Houri:
*Programming, Games and Transportation Networks*... (previous) ... (next): $1$. Preliminary ideas; sets, vector spaces: $1.1$. Sets - 1970: Arne Broman:
*Introduction to Partial Differential Equations*... (previous) ... (next): Chapter $1$: Fourier Series: $1.1$ Basic Concepts: $1.1.2$ Definitions