I'm trying to solve the following exercise.
Now, Rademacher's theorem says that locally Lipschitz functions are $\mathcal L^N$-a.e. differentiable, so $E$ must be a null set, and this is clearly the case. The suggested function is Lipschitz by the triangular inequality. I get stuck when I try to identify the "suitable subsets" of the sets $I_{j,n}$. We want select them in such a manner that upper and lower derivatives do not coincide at each point in $E$. For this it is enough that at each point at least one partial derivative does not exist. Does it suffice to fix an $N$ and take the union over $j$ and over $n$ running from $1$ to $N$? i.e.,
$$ E := \cup_{n=1}^N \cup_{j=0}^{2^{n-1}-1} I_{j,n} ?$$
If not, may you provide hints or a full solution?