Orders of roots of zero function

Question

In my course, I'm not given very precise definitions, so I'm relying on Wikipedia a bit.

https://en.m.wikipedia.org/wiki/Zeros_and_poles

1. Is order of roots/poles only defined for meromorphic functions?
2. What are the orders of the roots of the zero function, $f(z)=0$?

$f$ is holomorphic (and therefore meromorphic), but for all roots $z_0$ of $f$, there doesn't exist $n\in\mathbb{Z}$ such that $(z-z_0)^nf(z)$ is holomorphic and non-zero in the neighbourhood of $z_0$.

Answer 1

$f=0$ is the only analytic function with this property. If $f$ is analytic in a region $D$, $f(z)=0$ and $f$ is not the zero function then it has a zero of some finite order.