TITLE: Rademacher theorem
QUESTION [27 upvotes]: If $f:\mathbb{R}^n\to\mathbb{R}^m$ is of class $C^1$ and $\operatorname{rank} Df(x_o)=k$, then clearly $\operatorname{rank} Df\geq k$ in a neighborhood of $x_o$. It is not particularly difficult to prove the following counterpart of this result for Lipschitz mappings (very nice exercise). Recall that by the Rademacher theorem Lipschitz mappings are differentiable a.e.

Theorem. If $f:\mathbb{R}^n\to\mathbb{R}^m$ is Lipschitz, differentiable at $x_o$ and $\operatorname{rank} Df(x_o)=k$, then in
  any neighborhood of $x_o$ the set of points satisfying
  $\operatorname{rank} Df\geq k$ has positive measure.

This result is so natural that I am sure it has been observed before.

Question. Do you know a reference for this result?

I might use this result in my research, and I would prefer to quote it rather than prove it.

REPLY [6 votes]: While I don't have a reference, let me add another perspective. The following result is proved e.g. in the book Functions of Bounded Variation and Free Discontinuity Problems by Ambrosio, Fusco and Pallara (see the proof of Theorem 2.16).

Theorem. Let $n\ge k$. If $f_j\to f_\infty$ is a converging sequence of Lipschitz maps from $\mathbb R^n$ to $\mathbb R^k$ (pointwise or equivalently locally uniformly), then
$$ \det(\partial_{1}f_j,\dots,\partial_kf_j)\rightharpoonup\det(\partial_{1}f_\infty,\dots,\partial_kf_\infty) $$
in the weak*-$L^\infty$ topology.

Your observation can be seen as a nice corollary of the above theorem.
Indeed, assuming wlog that $m=k$ and $L(x):=\sum_{i=1}^k\partial_i f(x_0)x_i$ is an invertible linear map, you can apply the theorem to $f_j(\cdot):=r_j^{-1}f(x_0+r_j \cdot)$, for any sequence of radii $r_j\downarrow 0$, getting that
$$ \mathcal{L}^n(\{x\in B_1:\det(\partial_{1}f_j(x),\dots,\partial_kf_j(x))\neq 0\}>0 $$
eventually (since $f_\infty=L$), which trivially implies your claim.