I know that if a function $f$ is complex differentiable in a neighborhood of $z_0$, then we say it's holomorphic in $z_0$ and it's also analytic in a neighborhood of $z_0$.
But suppose that I know that $f'(z_0) = 0$ in the complex sense. I don't know what happens in a neighborhood of $z_0$; so can I say that the function $f$ is still analytic in $z_0$?
I don't think we can conclude that there is a neighborhood of $z_0$ such that the function is analytics, but then again I'm not sure.
Are there maybe additional condition to assume to ensure that the function is analytic?