Borel Structure of the Set Where a Partial Derivative Exists

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 $d_{E}$, topology and the notions of open and closed subsets of $\mathbb{R}^{n}$, and the Borel $\sigma$-algebra $\mathcal{B}(\mathbb{R}^{n})$, which contains every closed subset of $\mathbb{R}^{n}$, are as fixed there. Write $\mathbb{Q}$ for the rational numbers and $d_{\mathbb{R}}$ for the absolute value metric $d_{\mathbb{R}}(s,t)=|s-t|$ on $\mathbb{R}$.

Let $f:\mathbb{R}^{n}\to\mathbb{R}$ be continuous on $\mathbb{R}^{n}$, as a map from $(\mathbb{R}^{n},d_{E})$ to $(\mathbb{R},d_{\mathbb{R}})$. Let $i$ be a natural number with $1\le i\le n$ and let $e_{i}$ be the standard basis vector of $\mathbb{R}^{n}$ whose $i$th coordinate is $1$ and whose other coordinates are $0$, so that for $y\in\mathbb{R}^{n}$ and $s\in\mathbb{R}$ the point $y+se_{i}$ is obtained from $y$ by replacing its $i$th coordinate $y_{i}$ by $y_{i}+s$ and leaving the others unchanged. For $y\in\mathbb{R}^{n}$ and $s\in\mathbb{R}$ with $s\neq0$ put

$$\Delta_{s}(y)=\frac{f(y+se_{i})-f(y)}{s}.$$

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 $\partial_{i}f(y)$; by that definition, applied with the open set $\mathbb{R}^{n}$, membership $y\in E_{i}$ with value $L\in\mathbb{R}$ means precisely that for every $\varepsilon\in\mathbb{R}$ with $0<\varepsilon$ there is $\delta\in\mathbb{R}$ with $0<\delta$ such that $|\Delta_{s}(y)-L|<\varepsilon$ for every $s\in\mathbb{R}$ with $0<|s|<\delta$. For natural numbers $j$ and $k$ put

$$E_{j,k}=\Bigl\{\,y\in\mathbb{R}^{n}\;:\;|\Delta_{s}(y)-\Delta_{s'}(y)|\le\tfrac{1}{k}\ \text{ for all }s,s'\in\mathbb{Q}\text{ with }0<|s|\le\tfrac{1}{j}\text{ and }0<|s'|\le\tfrac{1}{j}\,\Bigr\}.$$

Then the following hold.

1. (Closedness) (i) For all natural numbers $j$ and $k$ the set $E_{j,k}$ is a closed subset of $\mathbb{R}^{n}$; in particular $E_{j,k}\in\mathcal{B}(\mathbb{R}^{n})$.

2. (Real increments) (ii) If $y\in E_{j,k}$ then $|\Delta_{s}(y)-\Delta_{s'}(y)|\le 1/k$ for all $s,s'\in\mathbb{R}$ with $0<|s|\le 1/j$ and $0<|s'|\le 1/j$.

3. (Exhaustion) (iii) $E_{i}=\bigcap_{k\in\mathbb{N}}\bigcup_{j\in\mathbb{N}}E_{j,k}$. In particular both $E_{i}$ and $\mathbb{R}^{n}\setminus E_{i}$ belong to $\mathcal{B}(\mathbb{R}^{n})$.

4. (Uniform approximation) (iv) If $y\in E_{i}\cap E_{j,k}$ then $|\Delta_{s}(y)-\partial_{i}f(y)|\le 1/k$ for every $s\in\mathbb{R}$ with $0<|s|\le 1/j$.