Check the Singularity of $f(z) = \tan\frac{1}{z-2}$ at $z=2$, where $z$ is a complex number.
What type of singularities $f(z)$ has at $z=2$? We know that Singularity is mainly of two types 1) Isolated 2) Non-isolated Again, Isolated singularity is of three types. i) Removable singularity ii) Pole iii) Essential Singularity.
Now, I found that the set of singular points of $f(z)$ is $$\left\{\frac{2}{(2n+1)\pi}+2 \, | \, n \in \mathbb{Z}\right\} \cup \{2\}.$$
Can it have both non-isolated and essential singularity at $z=2$ at the same time? It seems to be confusing to me.