The question of differentiability of a typical Lipschitz function inside a given analytic subset \(N\) of \(\mathbb{R}^d\), \(d \geq 2\) is studied. It is exposed a complete characterisation of the subsets \(N\) of \(\mathbb{R}^d\) in which a typical 1-Lipschitz function has points of differentiability: they cannot be covered by an \(F_{\sigma}\)-purely unrectifiable set (such sets are considered as typical differentiability sets). It is also shown that for all remaining sets \(N\), a typical 1-Lipschitz function is nowhere differentiable, even directionally, inside \(N\).
A set \(S \subseteq (0, 1)^d\) is considered as a typical differentiability set if a typical 1-Lipschitz function has points of differentiability in \(S\), i.e. Diff\( (f) \bigcap S\not= \emptyset\). Subsets of \((0, 1)^d\) in which a typical 1-Lipschitz function has no points of differentiability is considered as typical non-differentiability sets. A priori, a set \(S \subseteq (0, 1)^d\) may have exactly one of these two properties, or none.