Definition:Left-Total Relation/Multifunction/Branch

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Definition

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 $f$.


Principal Branch

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

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

Let $\sequence {S_i}_{i \in I}$ be a partitioning of the codomain of $f$ into branches.


It is usual to distinguish one such branch of $f$ from the others, and label it the principal branch of $f$.


Sources