>Let $\mathbb{D}^n$ be the closed $n$-dimensional unit ball, and let $f:\mathbb{D}^n \to \mathbb{R}^n$ be smooth.
Suppose that $df$ is invertible, except on a set of measure zero*.

Set $X$ to be the **union** of $\text{GL}^+_n$ and $$\{ A \in M_n \, | \,\det A \le 0 \, \text{ and the smallest singular value of $A$ has multiplicity } 1\}.$$

Here $M_n$ is the space of real $n \times n$ matrices, and 
$\text{GL}^+_n=\{ A \in M_n \, | \, \det A>0\}$. $X$ is an open dense subset of $M_n$.

>**Question:** Do there there exist $f_n \in C^{\infty}(\mathbb{D}^n, \mathbb{R}^n)$ such that $f_n \to f$ in $W^{1,2}(\mathbb{D}^n, \mathbb{R}^n)$ and $df_n \in X$ **everywhere** on $ \mathbb{D}^n $? 

I am fine with the $f_n$ being only $C^2$, if that matters.

*Comment:*

$X$ contains the space of matrices with distinct singular values, so if we can perturb $f$ so its differential has distinct singular values, we are done.
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** I am fine with assuming that $df$ is invertible outside a set of Hausdorff dimension not greater than $n-1$.