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

We work in the setting of Euclidean Space and Lebesgue Measure: Standing Notation, 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 A Derivative Matrix 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

Put ¶

Then $T(S)$ is $\lambda_{n}$-null. ¶