# 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 $\RR$ 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.

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### 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 $x$.

## Sources

- 1973: G. Stephenson:
*Mathematical Methods for Science Students*(2nd ed.) ... (previous) ... (next): Chapter $1$: Real Numbers and Functions of a Real Variable: $1.3$ Functions of a Real Variable: $\text {(a)}$*Many-valued Functions* - 1981: Murray R. Spiegel:
*Theory and Problems of Complex Variables*(SI ed.) ... (previous) ... (next): $2$: Functions, Limits and Continuity: Single- and Multiple-Valued Functions