My question arise from the study of the possible extensions of Rademacher's Theorem to the Sobolev Space $W^{1,p}(\Omega)$, with $\Omega\subset \mathbb{R}^n$. In specific I'm studying the proof of the fact that any element of $W^{1,p}(\Omega)$ is differentiable almost everywhere if $p>n$.
A key result in the proof is that any element in $W^{1,p}(\Omega)$, if $p>n$, has a continuous representative. Unfortunately I did not found any reference for this result.
Moreover, looking on Wikipedia's page on Sobolev inequalities, I discovered that in my setup (which is part of the general case $k<\frac{p}{n}$) every element of $W^{1,p}(\Omega)$, if $p>n$, should be an Holder's continuous function.
I'm puzzled by that, because I always thought to the element of a Soboloev Space as class of functions, and seems to me unrealistic that any element of any class of those Sobolev spaces is Holder continuous (which, if I'm not wrong, is stated as a property that holds everywhere).
So my questions are: 1) Do you have a reference which explain why any class of functions in $W^{1,p}(\Omega)$ (for p>n) has a continuous representative? 2) Is it correct the result stated by Wikipedia on Holder continuity? If the answer is yes, where am I wrong in thinking Sobolev functions?
Thank you very much for your time!