My professor claimed the following result:
If a function has an essential singularity at $z_0$
Then it can be approximated by $e^{1/z_0}$, or in general, $e^{f(z)}$, where $f(z)$ has a pole at $z_0$.
The only example of functions with an essential singularity that I know is $e^{1/z_0}$
I thought about applying Stone–Weierstrass Theorem, that exponentials and polynomials are dense in the space of continuous functions, but I am not sure how to rigorously prove the above claim.
Many thanks in advance!