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.
Best Answer
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.