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I tried to read as many notes and answers possible before asking this, but I find myself in need of some answer from you guys.

So my confusion is about multivalued functions VS set valued functions VS vector functions.

To say, an example of multivalued function is the cubic root, in $\mathbb{C}$ of some complex number.

An example of set-valued function might be the function that outputs the divisors of a given number.

An example of vector function might be $f(x, y) = x^2\hat{i} + 2y \hat{k}$ or $f(x) = x^2 \hat{i} + x \hat{k}$.

  • Question $1$: why isn't the square root in $\mathbb{R}$ considered a multivalued function? I mean, every number, except zero, has two roots...

  • Question $2$: reading on Wikipedia (in other languages), they highlight some difference between multivalued functions and vector functions. Why do they do this? I mean vector valued function are not multivalued functions, right? The fact that the outpus is a vector, doesn't mean there are multiple outputs for a given input...

  • Question $3$: According to what I understood, set-valued functions are not multivalued functions either. Is this correct?

In the end, I just would like to get if set-valued functions and vector functions are indeed not multivalued functions, or if not in general but yes in particular (that is, there are some example of set-valued / vector functions which are also multivalued).

Thank you and sorry!

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The standard convention is that "function" means what you call "single valued function".

It is true that every positive real number has two square roots. By convention, the square root function returns the positive root. For nonzero complex inputs there is no universally recognized way to single out one of the two roots. Then you can call the square root function "multivalued".

A vector valued function is an ordinary (single valued) function that happens to return a single vector for each input.

A set valued function is single valued. The value output is a single set. You could if you wished consider a multivalued function as an ordinary function that returns a set of outputs for each input. Whether you want to do that depends on context. The goal is always to choose vocabulary that helps the reader understand the underlying mathematical reality.

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  • $\begingroup$ That was really clear, thank you! $\endgroup$
    – Heidegger
    Commented Feb 27 at 20:11

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