# Definition:Real-Valued Function

## 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 do not use the hyphen: real valued function.

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

Sources which are primarily concerned with vector analysis may be seen to use the notation scalar valued function.