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:
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?
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) $\ \ \ \ f(x)=\pm\ x^2$
- 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?