# Uniqueness set for holomorphic function in $\mathbb{C}^n$

We know that if $$f:\mathbb{C}\to \mathbb{C}$$ is a holomorphic function and $$f$$ vanishes on a set $$E\subset \mathbb{R} \subset \mathbb{C}$$ such that $$E$$ has a limit point in $$\mathbb{R}$$ then $$f$$ is identically zero.

My question: Is the same is true in multivariable case? I.e. if $$f:\mathbb{C}^n\to \mathbb{C}^n$$ is a holomorphic function and $$f$$ vanishes on a set $$E\subset \mathbb{R}^n \subset \mathbb{C}^n$$ such that $$E$$ has a limit point in $$\mathbb{R}^n$$ then is $$f$$ identically zero?

$$f(z_1,z_2,...,z_n)=(z_1,z_1,...,z_1)$$ is a counterexample.