Branch Point

A point in the domain of a multivalued complex function where analytic continuation along closed loops around the point produces different values (nontrivial monodromy), so that a single-valued branch cannot be defined on any neighborhood that contains a loop encircling the point.

A point a in the complex plane around which analytic continuation of a multi-valued function fails to return to its original value, forcing multiple sheets or branch cuts; monodromy around the point is nontrivial.

A point in a covering map or projection where two or more local sheets meet and the local degree of the map drops; locally the map fails to be a homeomorphism (or is not locally bijective) because distinct preimages coalesce.