Let $X = \mathsf{Lip(\mathbb{R}^n;\mathbb{R}^n)}$ be the space of (globally, or perhaps locally) Lipschitz functions, say from $\mathbb{R}^n$ to $\mathbb{R}^n$. From Rademacher's theorem we know that $f \in \mathsf{Lip(\mathbb{R}^n;\mathbb{R}^n)} $ is almost everywhere differentiable.
What's the natural codomain $Y$ for the gradient (or jacobian) map $\nabla$?
$$ \nabla: \mathsf{Lip(\mathbb{R}^n;\mathbb{R}^n)} \rightarrow Y $$
(for example, if $X=\mathcal{C}^1(\mathbb{R})$, the "natural codomain" would be $Y=\mathcal{C}^0(\mathbb{R})$ due to the fundamental theorem of calculus - and I am not even sure about $X=\mathcal{C}^1(\mathbb{R^n})$)