If a holomorphic map from an open disc to $\mathbb{C}^n$ extends continuously to the closed disc, what about its partial derivatives? 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 $\overline{D}$?
Thanks

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. 
