Sobolev Embedding into $L^{\infty}$ I tried to find the question here, but I couldn't.
I'm a bit puzzled by Sobolev embeddings at the moment. 
In my lecture notes, I found the statement 

"If $\Omega \subseteq \mathbb{R}^d$ is bounded and open with $\partial \Omega \in C^1$, $1 \leq p < \infty$, $k,m \in \mathbb{N}_0$ with $m > k$, then the embedding 
  $$ W^{m,p}(\Omega) \hookrightarrow C^k(\overline{\Omega})$$ 
  is compact, if $m - \tfrac{d}{p} > k$."

In my case, I have $k = 0, \, p = 4, \, m = 1, d \leq 3$. 
So the first question is: 
$1.)$ From this theorem, I could conclude that functions in $W^{1,4}(\Omega)$ can be identified by functions in $C^0(\overline{\Omega})$, right? 
Because if I understand it correctly, this would imply, that I can always find a bounded representant in $W^{1,4}(\Omega)$.
But now I am puzzled, because the book where I wanted to refresh my knowledge about Sobolev Spaces says, that it is not possible to have a continous embedding of the form 
$$ W^{m,p}(\Omega) \hookrightarrow C^0(\Omega).$$
(The book gives with $d = 2$ the function $u(x) = \log(\log|x|)$ as example, but I didn't understand that completely yet.)
$2.)$ If this statement is true, why does the embedding work if I take the closure of $\Omega$? 
Thank you! 
 A: Since you have not received an answer yet, I thought I would share my thoughts on this.
The theorem you cite is correct. It´s statement and proof can be found in Adams, Sobolev Spaces, Theorem 6.2 Part III.
Furthermore, for $\Omega \subset R^{n}$ open, bounded by a sufficiently smooth boundary and $mp > n, \; j\ge0$
we have the (compact) imbedding:
$ W^{j+m,p}(\Omega) \subset \subset C_{B}^{j}(\Omega)$ - this is Part II in the above theorem.
I am not familiar with the counterexample you provided,
but I can assure you that for $p > n$ and sufficiently regular $\Omega$,
we have $ W^{1,p}(\Omega) \rightarrow C^{0,\gamma}(\Omega)$ - the latter being a Holder space.
A bounded set of Holder-continuous functions can always be extended (holder-continuousely) to the boundary of $\Omega$ - this of course is not true in general for continuous functions!
We can then use the Arzela-Ascoli-theorem to show compactness in $C^{0}(\bar \Omega)$, because in any such chain of (continuous) imbeddings, it is always sufficient that only one of them is compact for the hole chain to be compact.
I hope this helps a bit, take a look at the section in Adams and you will probably find more answers.
A: Thank you so much for your answer! 
I actually managed to resolve my confusion. 
When calculating the given example in the book, one comes to a point where the decisive question is: 
Is $$ \int_0^a \frac{r}{(r \log(r))^p} \, \mathrm{d} r < \infty$$
for $a <  1$. 
But this integral is only finite if $p \leq 2$, so actually the counterexample in the book holds only if $p \leq d$ without stating that at any point. 
