Name for a function that is defined for every point in the set that contains its domain? 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?
 A: 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


*

*there is some (necessarily unique) $y \in Y$ such that $f(x) = y$, or

*$f(x)$ is undefined.


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


*

*there is some $y \in [0,\infty)$ such that $y^2 = x$, and so $\sqrt{x} = y$, or

*there is no such $y$ (i.e. $x < 0$), and so $\sqrt{x}$ is undefined.


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 $X'\subsetneq Y$.
