# Definition:Real-Valued Function

## Definition

Let $f: S \to T$ be a function.

Let $S_1 \subseteq S$ such that $f \left({S_1}\right) \subseteq \R$.

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 see

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