In a paper by V Sverak (1988, Arch. Ration. Mech. Anal.; Example 1, p.119), the following function $u$ serves as the building block of an important counterexample:
''Consider a function $u \in H^{1}_{2,0}(G)$ that is essentially discontinuous at every point of G:=(-1,1) \times (-1,1)...''
Hoping to understand Sverak's counterexample, I have the following two questions:
(1) What does ''essentially discontinuous'' mean? (I think a plausible definition is that $u$ is essentially discontinuous at $x$ iff its representative $\tilde{u}(x):=\lim_{r\searrow 0} \frac{1}{|B(x,r)|} \int_{B(x,r)}f(y)\,dy$ either fails to exist, or is discontinuous, at $x$).
(2) With the understanding $H^1_{2,0}(G)=W^{1,2}_0(G)$, what is an example of $u$? Of course, there are standard examples to illustrate the failure of the embedding of $W^{1,2}(G)$ into $C^0(G)$, but I do not think they can be (easily) generalised to everywhere discontinuous functions.