---
tag: 03CPD1HGF3
kind: concept
name: Multivalued function
aliases: [multifunction, many-valued-function, multiple-valued-function]
msc: ["26E25"]
sources:
  - url: https://en.wikipedia.org/wiki/Multivalued_function
    locator: lead section
  - url: https://en.wikipedia.org/wiki/Multivalued_function
    locator: Concrete examples
same_as:
  wikidata: Q629085
  wikipedia: Multivalued function
---

## Summary

A multivalued function from a set $X$ to a set $Y$ assigns to each $x \in X$ a subset $F(x)$ of $Y$. For at least one $x$, the subset $F(x)$ has two or more elements.
In complex analysis, the main examples come from the inverses of functions that are not injective.
For example, each $z \neq 0$ has two square roots, and it has infinitely many [[logarithms|0X2FY491PF]] and [[arguments|0ZZRSX91QR]].

## Remarks

A [[branch|0SQ1F2BEDS]] of a multivalued function $F$ on a set $U$ is a function $f$ with $f(x) \in F(x)$ for each $x \in U$, usually continuous or holomorphic.
A [[principal branch|0YKY4M3KRQ]] is one branch that a standard convention chooses.
Riemann surfaces give a way to treat a multivalued function as one single-valued function.

## Explanations

### L1 human

The equation $w^2 = 4$ has two solutions, $2$ and $-2$, so the square root of $4$ has two values.
A multivalued function keeps all the values together, and a branch picks one value at each point.