MHB Multi-value Function: What & Why

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Why multi-valued function is a function?
 
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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".
 
I posted this question on math-stackexchange but apparently I asked something stupid and I was downvoted. I still don't have an answer to my question so I hope someone in here can help me or at least explain me why I am asking something stupid. I started studying Complex Analysis and came upon the following theorem which is a direct consequence of the Cauchy-Goursat theorem: Let ##f:D\to\mathbb{C}## be an anlytic function over a simply connected region ##D##. If ##a## and ##z## are part of...
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