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 defined as real-valued on $S_1$.

That is, $f$ is defined as real-valued on $S_1$ 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 iff it is real-valued over its entire domain.

Also see

 * Real function, in which the domain and codomain are both subsets of $\R$.