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Yes, since the morphism of sheaves $\mathcal{O}_{\mathbb{C}^n} \to \mathcal{O}_{M \cup N}$ is surjective, it is surjective on global sections by Cartan's theorem B. Thus, any holomorphic function on $M \cup N$ has a holomorphic extension to $\mathbb{C}^n$, in particular, this holds for the function $f$ which is $\equiv 0$ on $M$ and $\equiv 1$ on $N$.