In order to explain about the concept of weak derivatives, I plan to give examples on them. I already manage one example for the single-variable case, but I think it would be better if I can provide one for the multi-variable case. But I'm stucked.
Can you give me an example of multi-variable function which is not classically differentiable but weakly differentiable?
For example $\frac{1}{\|x\|}$ belongs to $W^{1,1}(B)$, $B$ is unit ball in $\mathbb{R}^3$, but it is not even continuous thus not differentiable. It is discussed here on wiki.