Why is a function defined to have only a single value at a point? I'm aware people have asked why a function cannot have more than one value before, on this site and on others, but I found none of the answers satisfactory, which mostly seemed to amount to "because it's defined that way", and did not discuss the reason for this definition. I do not understand the rationale behind this definition, it just seems to me a strange constraint on what otherwise was a very general notion of a "mathematical transform".
An argument could be raised that, under the accepted definition, certain properties we can conveniently say are true for all functions, won't be true for all functions under the alternative definition. For example,
$$\forall x,y.x=y\implies f(x)=f(y) \text{ for any function }f$$
But consider the converse of this same statement,
$$\forall x,y. f(x)=f(y)\implies x=y\text{ for any }\textbf{injective}\text{ function }f$$
We're already used to saying that some properties hold only for certain kinds of functions; it isn't too far a stretch to then assign a category to the functions which can produce only a single value, and say the first statement holds true for all functions of that category.
Is there then a logical reason for this special constraint or is it just this way due to convention or historical reasons?
 A: There are many "transforms" from one set to another, if you think of a transform as just a set of pairs $(x,y)$.
One of the most useful transforms is the one that corresponds to the intuitive idea that you have some kind of a "rule" that tells you an element of the codomain whenever you're given an element of the domain. The cost of a load of coal depends on the number of tons you buy. The distance you travel at a fixed speed depends on how long you drive. The position of a satellite in  orbit around the Earth depends on [some set of inputs].  In each case the input determines just a single answer. It's that idea that mathematicians have codified in the abstract definition of a function.
If you changed the definition of a function to include "multi-valued functions" no actual mathematical facts would change. But many of the most common and useful statements about functions would have to start

If $f$ is a function [that is single valued] then ...

which would complicate exposition.
The accepted definition is not

a strange constraint on what otherwise was a very general notion of
a "mathematical transform".

It's the choice we've made precisely because it makes dealing with the most common situation easiest.
