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Let $F$ be a holomorphic map from an open disc $D \subset \mathbb{C}^n$ to $\mathbb{C}^n$ and suppose $F$ extends continuously to $\overline{D}$. Do the maps $\partial F_i / \partial z_k$ extend continuously to $\overline{D}$?


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Start with $n=1$: How does $\sqrt{1-z}$ behave? – WimC Apr 4 '12 at 14:05
Thanks for the counterexample. – lenders86 Apr 4 '12 at 14:55
Uitstekend, @Wim. – Georges Elencwajg Apr 4 '12 at 16:22
up vote 0 down vote accepted

The partial derivatives do not necessarily extend continuously to the closed disc. WimC gave $F(z)=\sqrt{1-z}$ as a counterexample in one dimension. In any dimension, $F(z)=(\sqrt{1-z_1},\dots,\sqrt{1-z_n})$ has the same property: each partial fails to have a continuous extension, and is not even bounded.

A situation in which one might hope for boundary smoothness is when $F$ is a biholomorphic map onto a smooth domain. In one dimension there is a very satisfactory result called the Kellogg-Warschawski theorem. The situation in several variables is much more involved: see this brief overview.

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