The Degenerate-Derivative Values of a Locally Lipschitz Map of $\mathbb{R}^n$ Form a Null Set

We work in the setting of Product Measures and Lebesgue Measure on ℝⁿ: Standing Notation and Background, whose notation is fixed for every dimension and is used here with a natural number $n$ satisfying $1\le n$: the Euclidean norm $\lVert\,\cdot\,\rVert$, distance $d_{E}$, dot product and notion of openness on $\mathbb{R}^{n}$, the Borel $\sigma$-algebra $\mathcal{B}(\mathbb{R}^{n})$, Lebesgue measure $\lambda_{n}$ and its null sets, the constant $\sigma_{n}$ with $\sigma_{n}^{2}=n$, and the conventions on images $T(A)$ and on Lipschitz maps are all as fixed there. In addition, real $n\times n$ matrices act on $\mathbb{R}^{n}$ by the matrix-vector product.

Let $U\subseteq\mathbb{R}^{n}$ be open and let $T:U\to\mathbb{R}^{n}$ be locally Lipschitz. If $T$ is differentiable at $x\in U$, its derivative there is represented by the Jacobian matrix $DT(x)$ in the sense of The Derivative Is the Jacobian Matrix, and Is Unique; explicitly, for every $\varepsilon\in\mathbb{R}$ with $0<\varepsilon$ there is $\rho\in\mathbb{R}$ with $0<\rho$ such that

$$\lVert T(y)-T(x)-DT(x)(y-x)\rVert\le\varepsilon\,\lVert y-x\rVert\qquad\text{for all }y\in U\text{ with }\lVert y-x\rVert\le\rho .$$

Put (i)

$$S=\bigl\{x\in U: T\text{ is differentiable at }x\text{ and there is }\nu\in\mathbb{R}^{n}\text{ with }\lVert\nu\rVert=1\text{ and }\nu\cdot\bigl(DT(x)h\bigr)=0\text{ for every }h\in\mathbb{R}^{n}\bigr\}.$$

Then $T(S)$ is $\lambda_{n}$-null. (ii)