Is there a sample of a $f(x)=y$ multivalued function whose inverse $f(y)=x$ is also multivalued?

Question

Trying to learn about the properties of the multivalued functions, I found the definition at the Wikipedia as "a left-total relation (that is, every input is associated with at least one output) in which at least one input is associated with multiple (two or more) outputs".

I understand that, for instance $f(x)=\sqrt x$ in $\Bbb R$ is a multivalued function, as other functions applied to the complex numbers, and some trigonometric functions like arctan, arcsin, etc. (basically those are the samples explained at the wiki page).

Please I would like to share with you the following doubts:

1. Is there a sample of a $f(x)=y$ multivalued function whose inverse $f(y)=x$ is also multivalued, or multivaluation, if happens, is only possible in one direction?

2. Is it possible to define a multivalued function using a notation like this (e.g)? Is it possible to apply derivatives, integrals, etc?

(e.g) $\ \ \ \ f(x)=\pm\ x^2$

3. Apart from the basic examples that can be found at the Wikipedia (basically listed above), are there good examples of non-trivial multivalued functions useful for some field of Mathematics?

Thank you!

Answer 1

The inverse of $f(x)=\pm x^2$ would be $f^{-1}(x)=\pm\sqrt{|x|}$
In this case, there is a longer chain $\pm x \to |x| \to x^2 \to \pm x^2$, whose inverse is $\pm y\to |y| \to \sqrt{|y|}\to\pm\sqrt{|y|}$, so it is two single-valued functions in the middle, with domain and range $\mathbb{R}_{\geq0}$.

Answer 2

If you let $A$ and $B$ be sets, then the multivalued function that associates every $a\in A$ with every $b\in B$, then (provided $A$ and $B$ have more than 1 element) gives you an example of a multivalued function with multivalued inverse.

I would be careful with declaring that, for instance "$\sqrt{x}$ is multivalued". As it is normally interpreted, $\sqrt{x}$ returns the non-negative square root.

As for notation, you can define a function (multivalued or not) in any way you like, as long it is clear what you mean!

Even defining continuity would be odd here, maybe something like $\forall \epsilon >0, \exists \delta>0 (|x-x_0|<\delta \Rightarrow |f(x)-f(x_0)|)$ where you are allowed to pick any of the possible values for both $f(x)$ and $f(x_0)$?

Part of the reason that we use (single-valued) functions so much is because they are single valued! A multivalued function is just a relation so that every $x$ in the domain has a corresponding $y$ in the range. Relations are certainly used all over the place (e.g. equivalence relations, orders) but I don't see a special reason to single out these ones.