The room was unfrozen after the request in the Math Mods' Office:

How can i show that its not continuous at $z = 0$

Let's have a look at slightly simpler $1/\exp(z)$, I do not think there is much of a difference.

If I look only at complex numbers of the form $z=iy$ then $$\frac1{\exp z}= \frac1{\cos y + i\sin y}.$$

$$\lim_{y\to0} \frac1{\cos y + i\sin y} = 1 \ne 0.$$

So this function is not continuous at $z=0$.

Does this solve the problem @BAYMAX?

Yes,yes .. here also we can take $z^4= w$ and proceed..

yes

:P Hello

Is it me you're looking for?

Nah, it's cool

Just saying hi

It was meant as a reference to Lionel Richie song which contains the lyrics "Hello, is it me you're looking for?" But maybe I am too old, which is why the attempt to make a joke was lost.