It is well known that if the zeroes of an analytic (holomorphic) function $f:\Omega \to \mathbb{C}$ must be isolated unless the function is identically zero, where $\Omega$ is a domain in $\mathbb{C}$. Consequently, if $f$ and $g$ are analytic in a domain $D$ containing $\alpha$ and $\exists$ a sequence $\{z_n\}$ in $D$ such that $z_n \to \alpha$ as $n \to \infty$, $z_n \neq \alpha$ and $f(z_n)=g(z_n)$ for all $n$, then $f(z)=g(z)$ for all $z \in D.$

Are there any same or similar result for multivariable analytic function $f: \Sigma \to \mathbb{R}$? Here $\Sigma$ is a domain in $\mathbb{R}^n.$

For example, if $f,g: \Sigma \to \mathbb{R}$ are analytic functions satisfying $f(z)=g(z)$ for all $z \in \Sigma_1$ and for some open subset $\Sigma_1$ in $\Sigma$, then $f\equiv g$ on $\Sigma.$

Please let me know if you have any comment or reference for this question.

Thanks in advance!


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