Continuous representatives in Sobolev Spaces 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!
 A: The theorem you are looking for is called sobolev embedding theorem (see here for the 3rd google hit).
It does not show that every representative of a class (an element of $W^{m,p}$) is (hölder) continuous (when you think about it a little bit, this is clearly impossible, because you can alter these functions on a set of measure zero to no longer be continuous), but rather, that there is a function representing a class, which has the desired property.
A: 1) In any of the books on Sobolev spaces. References are on the wiki page you cited. 
2) Yes. As a toy example the space $W_2^1([0,1])$ can be taken here. Suppose $f\in W_2^1([0,1])$. Let $g(x)=\int_0^x f'(y)dy\ $. Then $g'(x)=f(x)$ and $f-g\equiv C$ a.e. in $[0,1]$. From the other hand $$|\Delta g(x)|^2=\left|\int_x^{x+\Delta x}f'(y) dy\right|^2\le \left( \int_x^{x+\Delta x}dy\right) \int_x^{x+\Delta x}|f'(y)|^2 dy\le |\Delta x| \|f\|_{W_2^1}.$$ So the representative $g+C$ of the function $f$ is Hölder continuous. In the general case the original approach of Sobolev was exactly the same: to obtain an integral representaion of a function and then to use integral inequalities.
