= 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 is
$$
\boxed{f\text{ holomorphic on }B(a,r)\setminus\{a\}}.
$$
The convention includes <removable singularities>, even when a suitable extension would eliminate the missing point.
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