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Let $F:\mathbb{D}\rightarrow\mathbb{D}^n$ be a proper holomorphic map and $n\geq2.$ I have the following questions:

  1. Does F extends in a nbhd of $\overline{\mathbb{D}}$ holomorphically?

  2. If not what about the case when $F$ is injective?

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No. Take for example $$F(z)=(z,g(z)),$$ where $g$ is any holomorphic function $\mathbb{D}\to\mathbb{D}$ that doesn't extend. This $F$ is both proper and injective.

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