Aren't functions in Sobolev spaces defined only upto a set of measure zero?

Question

Aren't functions in Sobolev spaces defined only upto a set of measure zero? I see some people on math.SE arguing all the functions of the form $f:\Omega\to\mathbb{R}$, where $\Omega$ is a convex open subset of $\mathbb{R}^N$, they say if $f\in W^{1,N+1}$, then by default $f$ is continuous inside $\Omega$. I agree $f$ cannot have jump discontinuities, but what about isolated discontinuities? especially given that functions in Sobolev spaces are defined only upto a set of measure zero? I am really confused between these contradicting definitions and facts.

Answer 1

Yes, Sobolev functions are in $L^p$ so they are not functions but equivalence classes $[f]$ of functions. However, if $p>N$ then Morrey's theorem says that there is a representative in the equivalence class $[f]$ of a Sobolev function which is Holder's continuous.