For an open set $\Omega \subset \mathbb{R}^N$ and $p \leq N$ we know that there are functions in $W^{1,p}(\Omega)$ which don't belong to $L^{\infty}(\Omega)$. We also know that for $p>N$ there is a continuous representative of every $u \in W^{1,p}(\Omega)$, that is, for every $u \in W^{1,p}(\Omega)$ there is a continuous function $\tilde{u}$ such that $u = \tilde{u} $ almost everywhere.
So the question is:
Could you provide me an example (for $p \leq N$) of a function $u \in W^{1,p}(\Omega)$ which does not have any continuous representative?
If $N > 1$ and $\Omega = B(0,1)$ then $u(x) = \log \log \left( 1 + \frac{1}{||x||} \right) \in W^{1,N}(\Omega)$, the function is continuous on $B(0,1) \setminus \{ 0 \}$ and unbounded at the origin and so does not equal a.e to any continuous function defined on $B(0,1)$. If $N = 1$ then any Sobolev function has a (locally) absolutely continuous representative.