Continuous embedding of $W^{d,1}(\Omega)$ into $C(\overline{\Omega})$ I'm trying to prove that for $\space \Omega \subset {\mathbb{R}}^{d}, \space$ $\Omega \in C^{0,1} \space$ there is a continuous embedding of the Sobolev space $W^{d,1}(\Omega)$ into $C(\overline{\Omega})$.
I was advised to prove it for cuboid domains first and then use the "fact" that lipschitz domain can be expressed as a countable union of disjoint open cuboids and possibly a zero measure set. However I have no idea where this "fact" comes from and I also don't see how to apply it.
I'll appreciate any help.
 A: As already mentioned above, the case for $\Omega \subset \mathbb{R}^{d}$ requires extension. However, the problem in full space is not difficult if you know approximation by convolution, though I imagine a good reference is probably hard to find. 
First assume $u \in C_{c}^{\infty}(\mathbb{R}^{d}).$
Then notice the equality ( which one can easily prove by repeating integration by parts in each variable)
$$ u (x_{1},\ldots, x_{d}) = \int_{-\infty}^{x_{1}}\int_{-\infty}^{x_{2}}\ldots \int_{-\infty}^{x_{d}} \frac{\partial^{d}u}{\partial x_{1}\partial x_{2}\ldots \partial x_{d}}(y_{1}, \ldots, y_{d}) \rm{d}y_{1}\rm{d}y_{2}\ldots \rm{d}y_{d},$$
which holds for arbitrary $(x_{1},\ldots, x_{d}) \in \mathbb{R}^{d}.$
Taking absolute values and taking the supremum over $(x_{1},\ldots, x_{d}) \in \mathbb{R}^{d},$ one obtains, 
$$ \left\lVert u \right\rVert_{L^{\infty}(\mathbb{R}^{d})} \leq \left\lVert \frac{\partial^{d}u}{\partial x_{1}\partial x_{2}\ldots \partial x_{d}} \right\rVert_{L^{1}(\mathbb{R}^{d})} \leq \left\lVert u \right\rVert_{W^{d,1}(\mathbb{R}^{d})}.$$
Now by density of $C_{c}^{\infty}(\mathbb{R}^{d})$ in $W^{d,1}(\mathbb{R}^{d})$ ( which one can easily show by taking convolutions with smooth mollifiers), the inequality above holds for any $u \in W^{d,1}(\mathbb{R}^{d}).$ Notice that continuity will also follow from this. To see that, again use the density and choose a sequence of function $ \left\lbrace v^{n} \right\rbrace_{n \geq 1} \subset  C_{c}^{\infty}(\mathbb{R}^{d})$ such that $$ \left\lVert u - v^{n} \right\rVert_{W^{d,1}(\mathbb{R}^{d})} \rightarrow 0 \text{ as  } n \rightarrow \infty.$$ By the inequality above, we have, 
$$ \left\lVert u - v^{n}\right\rVert_{L^{\infty}(\mathbb{R}^{d})} \leq \left\lVert u - v^{n} \right\rVert_{W^{d,1}(\mathbb{R}^{d})} \rightarrow 0 \text{ as  } n \rightarrow \infty.$$
Thus, $u$ is the uniform limit of a sequence of uniformly continuous functions $\left\lbrace v^{n} \right\rbrace_{n \geq 1}$ and thus must be continuous. 
Hope it helps. 
