Essential Singularity may or may not be Isolated

Theorem

Let $f$ be a complex function with an essential singularity at $z_0 \in \C$.

Then $z_0$ may or may not be an isolated singularity.

Proof

This theorem requires a proof. In particular: An example of each. Motivation: to remove confusion and establish this basic fact. The author of this page is infodumping an encyclopedia entry and cannot remember the slightest inkling of his Complex Analysis studies except for the fact that nothing is intuitive and everything needs to be investigated with great care. You can help $\mathsf{Pr} \infty \mathsf{fWiki}$ by crafting such a proof. To discuss this page in more detail, feel free to use the talk page. When this work has been completed, you may remove this instance of {{ProofWanted}} from the code. If you would welcome a second opinion as to whether your work is correct, add a call to {{Proofread}} the page.