Is the zero set of an analytic function closed nowhere dense?

Given a non-constant analytic function $f(x)$ on a domain $D \subseteq \mathbb{R}^n$. I want to prove that

$\mathcal{Z} = \{x \in \mathcal{D} |f(x) = 0 \}$

is closed nowhere dense.

I originally wanted to prove this by contradiction as follows using the identity theorem.

"Assume that $\mathcal{Z}$ is not a closed nowhere dense set, then $\mathcal{Z}$ must contain a limit point", which shows that $f(x)$ is constant zero by the identity theorem and thus contradicts that $f(x)$ is non-constant.

However, I am not sure if the following statement is true:

"Assume that $\mathcal{Z}$ is not a closed nowhere dense set, then $\mathcal{Z}$ must contain a limit point"

1. If it is true, could you provide one reference?

2. If not, is $\mathcal{Z}$ closed nowhere dense, and how can I prove it?