# Definition:Left-Total Relation/Multifunction

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## Definition

In the field of complex analysis, a left-total relation is usually referred to as a multifunction.

A multifunction may not actually be a mapping at all, as (by implication) there may exist elements in the domain which are mapped to more than one element in the codomain.

However, if $\RR$ is regarded as a mapping from $S$ to the power set of $T$, then left-totality of $\RR$ is the same as totality of this lifted function.

See the definition of a direct image mapping.

### Branch

Let $D \subseteq \C$ be a subset of the complex numbers.

Let $f: D \to \C$ be a multifunction on $D$.

Let $\family {S_i}_{i \mathop \in I}$ be a partitioning of the codomain of $f$ such that:

$\forall i \in I: f \restriction_{D \times S_i}$ is a mapping.

Then each $f \restriction_{D \times S_i}$ is a branch of $f$.

## Also known as

A multifunction is also known as a many-valued function, a multiple-valued function or a multi-valued function.

On $\mathsf{Pr} \infty \mathsf{fWiki}$ the terse form multifunction is preferred.

When the number of values is known to be $n$, the multifunction can be referred to as an $n$-valued function.

## Examples

### Arbitrary Multifunction

Consider the implicit function:

$y^2 = x + 2$

For $x > 2$, there are $2$ values of $y$ for every $x$.

Hence on that domain $y$ is a two-valued function of $x$.