The Partial Derivatives of a Lipschitz Function on $\mathbb{R}^n$ Exist Almost Everywhere

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 $1\le n$, and also with the dimension $1$: the Euclidean norm $\lVert\,\cdot\,\rVert$ and distance $d_{E}$, 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. As in A Lipschitz Function on an Open Interval is Differentiable Almost Everywhere we identify a point of $\mathbb{R}^{1}$ with its single coordinate and write $\mathbb{R}^{1}$ and $\mathbb{R}$ interchangeably.

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 $y,z\in\mathbb{R}^{n}$. Let $i$ be a natural number with $1\le i\le n$, let $e_{i}$ be the standard basis vector of $\mathbb{R}^{n}$ whose $i$th coordinate is $1$ and whose other coordinates are $0$, and let $E_{i}$ be the set of those $y\in\mathbb{R}^{n}$ at which the partial derivative of $f$ with respect to the $i$th variable exists, its value being written $\partial_{i}f(y)$. Then the following hold.

1. (Almost everywhere existence) (i) The set $\mathbb{R}^{n}\setminus E_{i}$ belongs to $\mathcal{B}(\mathbb{R}^{n})$ and is null.

2. (Bound by the Lipschitz constant) (ii) For every $y\in E_{i}$ one has $|\partial_{i}f(y)|\le L$.