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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?

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

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  • $\begingroup$ Sir thanking you for quick reply. Your counterexample pushed me to think whether the result holds for positive measure set in multivariable case. I asked that question in a different tab. Please give your thoughts. $\endgroup$ – Prof.Hijibiji May 7 '19 at 8:09

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