Multi-value Function: What & Why

[highmath]

Why multi-valued function is a function?

 

[S.G. Janssens]

You can always make a multi-valued function $f$ between sets $X$ and $Y$ single-valued by considering the associated function (in the narrow, traditional sense) that maps $X$ to the power set of $Y$.

In the context of set-valued analysis (which has many applications in e.g. microeconomics), the multi-valued functions are often called "correspondences". Some problems from the application domain can then be translated elegantly into questions about those correspondences. If you are interested, I can provide more references.

In other contexts, such as complex analysis, multi-valued functions often arise as inverses, and then one typically make a choice by convention and calls it the "principal value".

 

[math771]

A multi-valued function is still considered a function because it follows the fundamental definition of a function, which states that for every input, there is a unique output. Even though a multi-valued function may have more than one output for a single input, it still satisfies this requirement. Additionally, multi-valued functions are used in many mathematical and scientific applications, and they have been studied and proven to have important properties and characteristics. Therefore, they are considered valid and important functions in their own right.