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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) $\ \ \ \ f(x)=\pm\ x^2$

  1. 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!

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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 $\mathbb{R}_{\geq0}$.

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  • $\begingroup$ thank you! do you know other examples of (non-trivial, non-polynomic) two-ways multivalued functions? $\endgroup$ – iadvd May 8 '15 at 23:41
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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.

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  • $\begingroup$ thanks for taking time to answer. Your comment about continuity is very interesting! Regarding the special reason to single out these type of functions, for instance search here in MSE about "well-defined" versus "multivalued" functions, there are some good questions over there. :) $\endgroup$ – iadvd May 8 '15 at 23:34

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