Gradient Maps of a Function on an Open Subset of Euclidean Space

Let $d$ be a natural number, let $\lambda_{d}$ be Lebesgue measure on $\mathcal{B}(\mathbb{R}^{d})$, with its null sets, let $D\subseteq\mathbb{R}^{d}$ be open and let $f:D\to\mathbb{R}$. At a point $x\in D$ at which $f$ is differentiable, $Df(x)\in\mathbb{R}^{d}$ is the gradient of $f$ at $x$.

(Gradient map)¶ A gradient map of $f$ is a map $\nabla f:\mathbb{R}^{d}\to\mathbb{R}^{d}$, measurable with respect to $\mathcal{B}(\mathbb{R}^{d})$ on both sides, for which there is a null set $N\subseteq\mathbb{R}^{d}$ such that at every $x\in D\setminus N$ the function $f$ is differentiable and $\nabla f(x)=Df(x)$.