Complex function that is differentiable but not continuous in the neighborhood

Question

I'm looking for a complex function $f(z)$ such that it's differentiable at $z = z_0$ but isn't continuous in any neighborhood of $z = z_0$. I don't think differentiability in one point implies continuity in some neighborhood but I couldn't find a counterexample for that.

This question occurred to me when I encountered with the following theorem:

Answer 1

$f(x)=z^{2}$ for $z$ on the real axis and $f(z)=z^{3}$ everywhere else is such a function.