$z_0$ be a point of a domain of a complex valued function which is of isolated singularity.whether $z_0$ is a point of discontinuity of $f$?

Question

Let $f : D \to \mathbb C$ be a function and $z_0$ be a point of isolated singularity. Where $D$ is a domain.

My question is whether $z_0$ is a point of discontinuity of $f$ or not?

If not necessarily then please give some example.I have tried to find this question in stack exchange but failed.

Answer 1

Let $f(z)=\frac 1 z$ if $z\neq 0$ and $f(0)=1$. Then $0$ is an isolated singularity but $f$ is not continuous at $0$.

If $f$ is continuous at $z_0$ then $f$ cannot have an isolated singularity at $z_0$. It is automatically analytic at $z_0$.