Why aren't multi valued functions invertible?

Question

I recently learnt that functions are invertible if and only if they are bijective. But why aren't multi-valued surjective 'functions' invertible?

score 3 · Answer 1 · answered Oct 30 '13 at 06:51

A multivalued function is a misnomer because functions are by definition single-valued.

From Wikipedia:

Multivalued functions often arise from functions which are not injective. Such functions do not have an inverse function, but they do have an inverse relation. The multivalued function corresponds to this inverse relation.

score 2 · Answer 2 · answered Oct 30 '13 at 06:40

2

Because they are 'functions', not functions.

answered Oct 30 '13 at 06:40

Robert Israel

448,999

score 0 · Accepted Answer · answered Oct 30 '13 at 06:52

Noone would kill you if you wrote $f(x) = x^2$ and then $f^{-1}(4) = \{2, -2\}$ (You kind-of define your inverse as a set-valued function). But, forget about its linguistic content, "Invertible" is a technical term reserved for a specific purpose to signify a bijective function. You may say hello when leaving, and goodbye while coming, but that won't help much, will it?

Why aren't multi valued functions invertible?

3 Answers3