Rademacher's Theorem in $\mathbb{R}^n$

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 $\lVert\,\cdot\,\rVert$, dot product, distance $d_{E}$, notion of openness and closed balls, the Borel $\sigma$-algebra $\mathcal{B}(\mathbb{R}^{n})$, Lebesgue measure $\lambda_{n}$, the notion of a null subset, and the convention that Lipschitz maps between subsets of Euclidean spaces are understood for the restricted Euclidean distances, are as fixed there.

Let $L\in\mathbb{R}$ with $0\le L$ and let $f:\mathbb{R}^{n}\to\mathbb{R}$ be Lipschitz with constant $L$, that is $|f(y)-f(z)|\le L\lVert y-z\rVert$ for all $y,z\in\mathbb{R}^{n}$. Where all $n$ partial derivatives of $f$ exist at a point $x$, we write $Df(x)=(\partial_{1}f(x),\dots,\partial_{n}f(x))\in\mathbb{R}^{n}$ for the gradient of $f$ at $x$. Then the following hold.

1. (Differentiability almost everywhere) (i) There is a set $N\in\mathcal{B}(\mathbb{R}^{n})$ with $\lambda_{n}(N)=0$ such that for every $x\in\mathbb{R}^{n}\setminus N$ all $n$ partial derivatives of $f$ exist at $x$ and, for every $\varepsilon\in\mathbb{R}$ with $0<\varepsilon$, there is $\delta\in\mathbb{R}$ with $0<\delta$ such that

$$\bigl|f(x+h)-f(x)-Df(x)\cdot h\bigr|\le\varepsilon\,\lVert h\rVert\qquad\text{for every }h\in\mathbb{R}^{n}\text{ with }\lVert h\rVert<\delta .$$

2. (Identification of the derivative) (ii) At every point $x$ as in claim 1, the function $f$ is differentiable with derivative matrix the real matrix $A$ with one row and $n$ columns whose entries are $A_{1i}=\partial_{i}f(x)$ for $1\le i\le n$; by claim 2 of The Derivative Is the Jacobian Matrix, and Is Unique this matrix is the Jacobian matrix of $f$ at $x$.

3. (Bound on the gradient) (iii) At every point $x$ as in claim 1 one has $\lVert Df(x)\rVert\le L$.

4. (Locally Lipschitz maps into $\mathbb{R}^{m}$) (iv) Let $U\subseteq\mathbb{R}^{n}$ be open and let $T:U\to\mathbb{R}^{m}$ be locally Lipschitz on $U$. Then the set of those $x\in U$ at which $T$ is not differentiable is null. At every $x\in U$ at which $T$ is differentiable, the derivative matrix of $T$ at $x$ is the real matrix with $m$ rows and $n$ columns whose entry in row $k$ and column $i$ is the partial derivative $\partial_{i}T_{k}(x)$ of the $k$th coordinate function of $T$.