By definition, a function is defined for every value in its domain. However, in some cases there is a larger set that contains its domain, and it is natural to distinguish between functions whose domain covers it and those that don't. For example, $\sqrt x$ is defined for $x\geq 0$ which is a subset of $\mathbb R$, but $x^2$ is defined for all $x \in \mathbb R$. Is there a word for functions that "cover everything," relative to the universe of discourse, that distinguishes them from functions whose domain is a proper subset of the universe of discourse?
Name for a function that is defined for every point in the set that contains its domain
functionsterminology
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You're looking for partial function. A partial function $f$ from $X$ to $Y$ is a function $X' \to Y$, where $X'$ is some subset of $X$.
However, regarding your comment about functions that are not well-defined: there is no such thing as a function that isn't well-defined. If $f$ from $X$ to $Y$ is not well-defined, then $f$ is not actually a function, but only a relation (some subset of $X \times Y$). Partial functions are not considered functions either. The term "function" requires that for every input there is exactly one output.
What probably confused you is that we often define a function, and right after defining it check that it is well-defined. What we are really checking though, is that what we claimed was a function was in fact a function; that's why we call it well-defined, meaning our definition was not faulty. It is somewhat of an abuse of words to define something as a function before checking that it is well-defined; one should really first define it as a relation, and then prove the proposition that it is a function.
From the point of view of set theory, a function $f$ from $X$ to $Y$ is a subset of the Cartesian product $X\times Y$ which satisfies the property that $$ (x,y) \in f \ \text{and}\ (x',y) \in f \implies x=x'. $$ In essence, a function is a subset of the Cartesian product which satisfies the "vertical line property".
However, we typically don't like to think of functions in this manner. Instead, we like to think of functions as "arrows" which take an input from some domain and give an output in some codomain, perhaps according to some formula or rule. From this point of view, a function is defined by three data:
- a domain $X$,
- a codomain $Y$, and
- a rule which assigns elements of $X$ to elements of $Y$.
In describing a function in plain English (rather than notation), we might want to emphasize either the domain or the codomain, or properties of either of these sets.
If we want to emphasize some property of the codomain, we say that $f$ is a "$[Y]$-valued function". This means that the codomain of $f$ is either $[Y]$ where $[Y]$ is some set, or that the codomain of $f$ has some property $[Y]$. For example, a "vector-valued function" is a function whose codomain is a vector space, while an "$\mathbb{R}$-valued function" is a function with the real numbers as its codomain (such a function may also be described as "scalar valued" in many contexts).
If we want to emphasize some property of the domain, we typically just say that $f$ is "a function on $[X]$", where $[X]$ is a set. If the actual domain is not that important, but we only care about some property of that domain, we might say that $f$ is "a function on [some kind of space]", e.g. $f : \mathbb{R}^n \to \mathbb{R}$ could be described as a function on a vector space, or "a function of a vector variable". That being said, we generally like to specify the domains of functions fairly exactly, so I would think that such a vague description of the domain of a function would be uncommon.
In the cases raised by the question,
- a function $f : \mathbb{R}^m \to \mathbb{R}^n$ is a vector-valued function on $\mathbb{R}^m$, or perhaps a vector-valued function of a vector variable, or even a vector-valued function of multiple variables (if we don't care so much about the vector space structure of $\mathbb{R}^m$);
- a function $f : \mathbb{R}^m \to \mathbb{R}$ is a scalar-valued function on $\mathbb{R}^m$, or maybe a scalar-valued (or real-valued) function of a vector variable; and
- a function $f : \mathbb{R} \to \mathbb{R}^n$ is a vector-valued function on $\mathbb{R}$, or a vector-valued function of a scalar variable, or perhaps simply a scalar function of several variables.
The various phrases used above can be mixed-and-matched a little, and are all relatively informal. They should be easily understood by most readers (particularly in the kinds of contexts where they are used, which is probably a bit tautological, but... oh, well), but can and should be more explicitly defined if there is any potential for ambiguity.
Best Answer
Unless otherwise noted, the notation $$ f : X \to Y $$ indicates that $f$ is a function with domain $X$ and codomain $Y$. This notation is agnostic with respect to the existence of a larger set $X'$ such that $X \subsetneq X'$. That is, with respect to the usual definitions of a function, there is no notion or notation for a function which "covers everything" in some set—the domain of a function is precisely the set on which that function is defined; no more, no less.
However, if this distinction is important, then it may be appropriate to discuss partial functions. From a set theoretic point of view, a partial function from $X$ to $Y$ is a subset of the Cartesian product $X\times Y$ which satisfies the property that $$ (x,y) \in f \land (x',y') \in f \implies y=y'. $$ Equivalently, a partial function from $X$ to $Y$ is a function from $X'$ to $Y$, where $X' \subseteq X$. If $f : X \not\to Y$ is a partial function and $x \in X$, then either
Note that I have used the notation $X \not\to Y$ in order to differentiate between a function (in the usual sense) and a partial function.
For example, $$ \sqrt{\cdot} : \mathbb{R} \not\to \mathbb{R} $$ is a partial function, as the square root function is not defined for negative values. Here, if $x \in \mathbb{R}$, then either
The function $$ (\cdot)^2 : \mathbb{R} \not\to \mathbb{R}$$ is also a partial function, as the domain of this function is $\mathbb{R}$, and $\mathbb{R}\subseteq \mathbb{R}$. In a case like there—that is, if $X = X'$—then the function is called a total function.
While I am not aware of this use, I think that it would be reasonable to say that a partial function $f : X \not\to Y$ is a proper partial function if the set of $x \in X$ such that $f(x)$ is undefined is non-empty. In the notation used above, a proper partial function $f : X \to Y$ satisfies the property that $f$ is a function from $X' \to Y$, where $X'\subsetneq Y$.