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 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.
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
- For a video presentation of the contents of this page, visit the Khan Academy.