 Solution

A function has an isolated singularity at $a$ if, for some $r>0$, it is defined and holomorphic on the punctured disc $0<|z-a|<r$, while a holomorphic extension through $a$ has not been assumed. In particular, its domain omits $a$, or its assigned value there fails to give such an extension. The key local condition isThe convention includes removable singularities, even when a suitable extension would eliminate the missing point.