TITLE: Weakly differentiable but classically nowhere differentiable
QUESTION [10 upvotes]: Is there any example of a function which is weakly differentiable but none of its versions are classically differentiable anywhere (or differentiable only on a set of measure 0) ? Thanks

REPLY [7 votes]: In any Sobolev space where functions are capable of being infinite at a point, they are capable of being nowhere continuous.  
Let $f$ be a function in some Sobolev space which is infinite at 0 (for example $f$ might be $|x|^{-1/3}$ in $H^1(\Omega)$ for $\Omega$ some open ball in $\mathbb{R}^3$), let $q_n$ be an enumeration of the rational points of $\Omega$, and let $f_h(x) = f(x-h)$.  Then we can define $$g = \sum_n 2^{-n} f_{q_n}.$$  This $g$ is unbounded on any open set, but its norm is no more than double the norm of $f$, so in particular it is still weakly differentiable.  
One can easily verify that the sequence of partial sums converges, and that the result is nowhere continuous, [and that you can't make it continuous by modifying on a set of measure zero].