Let $D$ denote the unit disk in $\mathbb C=\mathbb R^2$. We consider the function $f:D\rightarrow\mathbb C $ defined by $$f(z):=\frac{z^2}{|z|}.$$ Then as proved in Global invertibility (p324 Remark 4), there doesn't exist a sequence of $C^1$ immersions $\{f_n\}$ such that $f_n\rightarrow f$ in $C^0(\bar D)$. However, I'm interested in the following case:
if we regard $f$ as a function from $D$ to $\mathbb R^3$ (by canonically embedding $\mathbb C$ into $\mathbb R^3$), can we approximate $f$ by a sequence of $C^1$ immersions in $C^0(\mathbb D)$?
The previous argument fails in this case.
