Locally Lipschitz Map on a Euclidean Open 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 natural numbers $n,m$ satisfying $1\le n$ and $1\le m$: the Euclidean norm $\lVert\,\cdot\,\rVert$, distance $d_{E}$, notion of openness, and closed balls $\bar{B}(x,r)$ are those fixed there, as is the convention that Lipschitz maps between subsets of Euclidean spaces are understood for the restricted Euclidean distances.

Let $U\subseteq\mathbb{R}^{n}$ be open and let $T:U\to\mathbb{R}^{m}$ be a map.

Definition. (i) The map $T$ is locally Lipschitz on $U$ if for every $x\in U$ there are $r,L\in\mathbb{R}$ with $0<r$ and $0\le L$ such that $\bar{B}(x,r)\subseteq U$ and the restriction of $T$ to $\bar{B}(x,r)$ is Lipschitz with constant $L$ as a map from the metric space $(\bar{B}(x,r),d_{E})$ to $(\mathbb{R}^{m},d_{E})$; that is,

$$\lVert T(y)-T(z)\rVert\le L\,\lVert y-z\rVert\qquad\text{for all }y,z\in\bar{B}(x,r).$$

The number $L$ is called a local Lipschitz constant for $T$ on $\bar{B}(x,r)$. (ii)