A real hyperplane of $\mathbb{C}^{n}$ is given by a equation $\textrm{Re}( \ell(Z)) = c$, where $c \in \mathbb{R}$ and $\ell(Z) = \sum_{j=1}^{n} a_{j}z_{j}, a_{j} \in \mathbb{C}$. How to prove that if $f$ is a holomorphic function of $\mathbb{C}^{n}$ that vanishes in a real hyperplane, then $f$ is identically zero?
Thank you!