Branch cut

A curve removed from the domain to make a chosen branch continuous everywhere else.

A branch cut is a curve (often a ray or a line) removed from the plane so that a chosen branch of a multivalued function is continuous at every remaining point. Without the cut, a branch is forced to jump discontinuously somewhere, because tracking the output continuously all the way around any loop enclosing the origin returns a different value than the one you started with; deleting a cut that blocks every such loop is what prevents that jump from ever being crossed.

Examples

The branch cut along the negative real axis, $(-\infty,0]$ , is the standard choice for the principal branch of $\arg z$ and $\log z$ .
Choosing a different branch cut, such as the positive real axis, produces a different (but equally valid) branch of the logarithm.