Exercise 15, § 3

Let $n\geqslant 1$ be an integer and $p$ a real number such that $1\leqslant p < n$. a) Let $\alpha$ be a real number such that $0< \alpha < n/p-1$. Let $g$ be a function on $\mathbf{R}_+^*$, indefinitely differentiable, positive, such that $g(x) =x^{-\alpha}$ for $x$ sufficiently small and $g(x) = 0$ for $x$ sufficiently large; the function from $\mathbf{R}^n$ into $\mathbf{C}$ such that $x\mapsto g(\|x\|)$ belongs to $W^{1,p}(\mathbf{R}^n)$.

b) Let $i\mapsto q_i$ be a bijection of $\mathbf{N}$ onto $\mathbf{Q}^n$ and let $f$ be the mapping from $\mathbf{R}^n$ into $\mathbf{C}$ defined by

for $x\in \mathbf{R}^n$. The series defining $f$ converges in $W^{1,p}(\mathbf{R}^n)$; for every open set U of $\mathbf{R}^n$, the restriction of $f$ to U is not differentiable almost everywhere. c) There exists a function $f\in W^{1,n}(\mathbf{R}^n)$ which is continuous at no point. (Similar method with the function $x\mapsto$ log($|$log($|x|$)$|$)$.$)