Branch point

A point around which a multivalued function cannot be split into consistent single-valued branches.

A branch point of a multivalued function is a point such that, no matter how small a punctured neighborhood is drawn around it, the function cannot be separated into consistent single-valued branches there -- continuing any one branch once around the point returns a different value. A branch cut is exactly the device used to avoid this: it is a curve, run between branch points (or from a branch point out to $\infty$ ), that blocks every loop from encircling a branch point alone.

Examples

$z=0$ is a branch point of $\sqrt z$ and of $\log z$ , since looping once around it lands on a different branch.
A branch cut for $\sqrt z$ must run from the branch point $z=0$ out to $\infty$ , also a branch point, rather than stopping partway.