Definition:Real-Valued Function

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 range is the set of real numbers $$\R$$.

That is $$f$$ is real-valued iff it is real-valued over its entire domain.