# Definition:Left-Total Relation/Multifunction

## 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 $\mathcal R$ is regarded as a function from $S$ to the power set of $T$, then left-totality of the relation 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.

## Sources

- 1981: Murray R. Spiegel:
*Theory and Problems of Complex Variables*(SI ed.) ... (previous) ... (next): $2$: Functions, Limits and Continuity: Single- and Multiple-Valued Functions