Is this definition of a function wrong?

Question

I am reading Complex Variables by Churchill and the author defines the generalization of a function in the following way:

A rule that assigns more than one value to a point z in the domain of definition.

This statement is giving me some trouble because I interpret it as $w\neq u \implies f(z)=w\,\wedge\, f(z)=u$ which is not the definition of a function. How am I supposed to interpret the above statement?

Answer 1

There is not much context to deduce what he means. Anyway, a function is a rule that assigns an object to an object. If you have two sets $X,Y$, then what the author probably meant is that you can define functions of the form $$f :X\to 2^Y\equiv P(Y), $$ the power set of $Y$. A typical example is the following. Take a function $$ f:\mathbb{R}\to\mathbb{R}. $$ If the function is injective, then its inverse is well defined. Otherwise it still makes sense to consider a multivalued inverse $f^{-1}:\mathbb{R}\to 2^\mathbb{R}$ in the following way $$ f^{-1}(y) = \{y\in \mathbb{R}: f(x)=y\}. $$ Take $f=x^2$ and $g=x^3$. $g^{-1}(x)=x^{1/3}$, while $$ f^{-1}(x) = \{+ \sqrt{x}, -\sqrt{x}\}, \quad x\geq 0. $$

Answer 2

As the comments to your question remark, in complex analysis, a multivalued function $f$ on the complex plane $\Bbb C$ is not a function, but rather a relation that, given $z\in\Bbb C$, offers a range of choices for the value $f(z)$. There is no fixed way to decide which of the choices is appropriate; it depends on the context. One way of fixing a choice is to "cut" the complex plane. If we want to retain continuity, we may find ourselves having to hop from one choice on the list to another.

The way to make rigorous sense of such "functions" is to replace the domain or range $\Bbb C$ by a manifold appropriate to the function. This is something like a multi-level car park which pancakes down onto $\Bbb C$.