1) Is there sequences of continuous function on $[0,1]$ that converge converge to a function that is nowhere continuous ?
2) Same question on all $\mathbb R$.
I was looking for a sequence that converge to $1_{\mathbb Q\cap [0,1]}$, but not conclusive.
As you can see in this discussion, the number of discontinuous points can be dense in $[0,1]$, but the set of discontinuous points must be a meagre set by the Baire-Osgood theorem. As a subset of a meagre or a countable union of meagre sets are also meagre (thank you wikipedia), I suppose $\Bbb R$ and $[0,1]$ are not meagre, otherwise this concept would be useless. So no, it is not possible.