A holomorphic funtion on an open connected domain, with $f(z_0)=0$

Question

Let $f$ be a holomorphic funtion on an open connected domain $\Omega \subseteq \mathbb{C}.$ Suppose $f$ is not the zero function. If $z_0$ is a zero of $f$, there exists a neighborhood $U \subseteq \Omega$ of $z_0$ such that $f(z)\neq0$ for all $z \in U \setminus \{z_0 \} $.

In these posts they do it using the series expansion:

• All the zeroes of analytic $f$ in $A$ are isolated, or $f \equiv 0$ on $A$.

• Complex analytic functions and their zeros.

But I was told to use the principle of analytic continuation; and I have not clue how to use it here.