What do we call a "function" which is not defined on part of its domain? Before the immediate responses come in, I realize that a properly defined function means that it is defined for every value in its domain.
My question is this: if $f:A\to B$ has the property $f(a)=b_1$ and $f(a)=b_2$, then it is often still called a function, but one which is "not well-defined".
If there is $b$ in $B$ such that there is no pre-image under $f$ then we say $f$ is "not surjective".
So what would we call a "function" which has the property that $f(a)$ is not defined for some $a$ in $A$? It seems like there should be a word for this, other than just saying $f$ is not a function.

Edit: I realize that a function which is not well defined is not actually a function. I'm talking about informal speak, for example in class how we say "let's check if this function is well defined" as though it were a function even if it weren't well defined. I'm wondering if there is an analogous phrase for maps which aren't defined on their whole domain. This is all informal, which is why I tagged it a soft question.
 A: Its not completely clear what you're looking for, but:

Definition 0. Let $Y$ and $X$ denote sets. Then a relation of type $Y \leftarrow X$ is, by definition, a subset of $Y \times X$.

Relations form a locally-posetal dagger category that is sometimes denoted $\mathbf{Rel}$.
Explicitly:
Given relations $q : Z \leftarrow Y$ and $p : Y \leftarrow X$, their composite is defined as follows:
$$(q \circ p)(z,x) \iff \mathop{\exists}_{y:Y}(q(z,y)\;\&\; p(y,x))$$
Given a relation $r : Y \leftarrow X$, its converse $r^\dagger : X \leftarrow Y$ is defined as follows:
$$r^\dagger(x,y) \iff r(y,x)$$

Definition 1. Suppose $r : Y \leftarrow X$ is a relation. Then:
  
  
*
  
*$r$ is said to be entire iff for all $x \in X$, there is at least one $y \in Y$ with $r(y,x)$.
  
*$r$ is said to be deterministic iff for all $x \in X$, there is at most one $y \in Y$ with $r(y,x)$.

These can be characterized in a "quasi-algebraic" manner:


*

*$r : Y \leftarrow X$ is entire iff $r^\dagger \circ r \geq \mathrm{id}_X$

*$r : Y \leftarrow X$ is deterministic iff $r \circ r^\dagger \leq \mathrm{id}_Y$
A deterministic relation is also called a partial function. Entire relations are sometimes called multifunctions. So a "function" that gives multiple outputs $b_1,b_2$ at some input $a$ can be referred to as a multifunction.
A: 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.
A: It is not possible that $f(a) = b_1$ and $f(a) = b_2$ unless $b_1=b_2$. However, we can consider functions that output sets, so $f(a) = \{b_1,b_2\}$ and $f(c) = \{b_3\}$. If we did that we could think of $f$ as taking multiple values on $a$ but only a single value on $b$. This would be called a multivalued function (https://en.wikipedia.org/wiki/Multivalued_function). E.g. inverses of trigonometric functions.
A: $\text{Partial function}$            - when some items in the domain map to nothing (for example $f:\mathbb{R}\rightarrow\mathbb{R}$ with $f:x\mapsto\frac{1}{x}$ is technically a partial function because it's undefined at $0$)
A: What do we call a “function” which is not defined on part of its domain?
The word "relation" is common for that sort of thing.
However, "relations" include non-functions in addition to function which are not defined on part of their own domain.
DEFINITION OF RELATION

For any set $R,$
     $R$ is a "relation"
         if and only if
      there exists a set $A$ such that $R \subseteq A \times A$

A NOTE ABOUT THE DEFINITION
If you want the set of inputs to a relation (its domain) and the set of outputs (co-domain) to be different sets, you can make that happen.

There exist sets $S$ and $T$ such that $R \subseteq S \times T$
  if and only if
there exists a set $A$ such that $R \subseteq A \times A $

Simply consider the union of sets $S$ and $T$. Take $A$ to be $S \cup T$
A SIMPLE EXAMPLE OF A FUNCTION WHICH IS NOT DEFINED ON PART OF ITS DOMAIN
Consider $F = \{(1, 11), (2, 22), (3, 33)\}$
You could also write $F$ like this:
$ \forall x \in \{1, 2, 3\}, \quad F(x) =
\begin{cases}
11,  & \text{ if } x = 1 \\   
22,  & \text{ if } x = 2 \\ 
33,  & \text{ if } x = 3 \\  
\end{cases}$
$F$ is a definitely a "relation".
$F$ is a function on the domain $\{1, 2, 3\}$
However, $F$ is NOT a function on the domain $\mathbb{N}$
$F$ is an example of a function which is not defined for some parts of its domain $\mathbb{N}$
What Properties Make a Relation a Function?
I like to wear the hat of a graph theorist.
I say that every element of the domain is a node that that the "degree" of $x$ is $1$ if there is exactly one line (or "edge") coming out of node $x$
There are four properties of interest:

*

*The degree of every node in the domain is at most one. (function)

*The degree of every node in the domain is at least one.  (function)

*The degree of every node in the co-domain is at most one. (injective)

*The degree of every node in the co-domain is at least one. (sujective)

A function is a relation such that the degree of every node in the domain is exactly equal to one.
Beware of Pitfalls
Some Warnings
Most mathematicians are very sloppy when writing about the following topics:

*

*relations

*functions

*injections

*domains, range (image), co-domain

*surjections

*bijections

*correspondences

*etc...

The phrase “the domain of a relation” could be used to describe at least two different sets.
Also, the way the word "domain" is used is not usually consistent with the formal definition of "domain" provided earlier in the paper.

A lot of mathematicians will say that, for any sets $A$ and $B$, if $R$ is a subset of $A \times B$, then $A$ is a domain of $R$.
In fact, mathematicians will often write that $A$ is "the" domain of $R$, even though $A$ is not unique.
From that definition of relation, you can prove absurd things.
For example, you can prove that the number $42$ is in the domain of every relation.
Proof:

Let $A$, $B$, and $R$ be sets such that $R \subseteq A×B$.
By the definition of relation, $R$ is a relation on $A×B$.
Note that $R⊂A×B$ implies that $R⊂(A∪\{42\})×B$.
By the definition of relation, $R$ is a relation on (A∪{42})×B.
By the definition of domain, any element in the set $A∪\{42\}$ is in the domain of relation $R$.
Therefore, $42$ is in the domain of relation R.
$\mathtt{QED}$

If you define a relation in such a way that it can be proved that every relation has $42$ in its domain, then you have a very silly definition of relation.


*

*Type One Domain

*

*INFORMAL DEF. We define the domain of a relation to be the set of numbers which appear on the left of at least one ordered pair in that relation.

*EXAMPLE if $F$ = { (1, 11), (2, 22) } then the domain of $F$ is set containing $1$, $2$, $3$ and nothing else.

*PROS One nice thing about this definition of "the domain" is that if I give you a set of ordered pairs (a relation) then there is only one logical choice for what the domain of the relation could be.

*CONS One bad thing about this definition of domain is that everything in the domain is guaranteed to "be hit". In many cases, we want relations and domains where not every element of the domain gets mapped.

*FORMAL DEFINITION: for any set $A$ and for any relation $R$, if $R \subseteq A \times A$ then, the domain of relation $R$ is ${\{x \in A: \text{ such that } \exists y \in A \text{ such that } (x, y) \in R\}}$



*Type Two Domain

*

*INFORMAL DEF. We define the domain of a relation to be a very specific super-set of the set of left-hand sides.

*EXAMPLE if $F$ = { (1, 11), (2, 22) } then we  might choose for the domain of $F$ to be the set $\{1, 2, 3, 4, 5\}$.

*CONS You cannot tell what the domain is by looking at a relation. A relation is a set of order pairs, right? Well, if I hand you a relation, what is the domain? The information about what the domain is, is not stored, or built-into, the relation itself. You can't know what the domain of the relation is unless someone tells you.



The Wikipedia articles on relations, injections, and surjections are not any better than published papers and textbooks.
If a relation is truly a set of ordered pairs then exactly one of the following is true:

*

*The domain of a relation cannot be determined from the relation alone.

*The domain of a relation can be determined only the relation itself, but everything in the domain is guaranteed to be hit.

One potential fix is to define a relation (or a function) as a graph $G = (VS, ES)$ such that $ES \subseteq VS \times VS$.
But then we have other problems.
A "free-floating node" is a dot having no lines coming out of it.
They are also called "isolated nodes" or "vertices of degree zero."
$$\{x: \forall y, (x, y) \not\in R\} \cup \{y: \forall x, (x, y) \not\in R\}$$
If we have a free-floating node (vertex or dot), then is the free-floating node an element of the domain (left-hand-side) or the co-domain(right-hand-side)?
Some mathematicians think that the canonical definitions for function and domain are elegant, but the formalisms are really a mess.
For example, we should have at least 4 words for the different types of domain, but we only have three words:





Description
Canonical Nomenclature




1
the set of $y$ such that $(x, y)$ in function $f$
the range (image) of $f$


2
a super-set of the set of $y$ such that $(x, y)$ in function $f$
the co-domain of $f$


3
some super-set of $x$ such that $(x, y) \in R$
the domain of $f$


4
the set of $x$ such that $(x, y) \in R$
????




there is no word for the set of $x$ such that $x$ is related to something.
Or, maybe, the word is domain.
But then, the word domain refers to two different things.
What?!?!?
