Can a meromorphic function $f:U \to \mathbb{C}, \; U\subset \mathbb{C}$ domain, that is not the zero function, have a zero of order infinity?

Question

Can a meromorphic function $f:U \to \mathbb{C}, \; U\subset \mathbb{C}$ domain, that is not the zero function, have a zero of order infinity?

(Clear: Would $f$ be holomorphic on the whole domain U then there is no zero of order infinity if $f\neq 0$)

Answer 1

Since zeros of a meromorphic function are isolated (cf. wikipedia), we can restrict $U$ to a small ball $B$ on which $f$ is holomorphic. Then, as Thomas Andrews pointed out in the comments, the identity theorem implies that an infinite order zero can only occure for the 0-function on $B$. By analytic continuation $f$ was zero on all of $U$.

Nonzero meromorphic $f$ thus connot have zero of infinite order.