Do Lipschitz-continuous funcions have weak derivatives on bounded open sets? Let $\Omega\in\mathbb{R}^n$ be open and bounded. I'm wondering if a function 
$f\in C^{0,1}(\Omega)$ (a Lipschitz-continuous one) is also an element of $W^{1,2}(\Omega)$ (that is the space of weakly derivatives functions whose first weak derivatives are $L^2$-functions). 
One can easily show that $\|f\|_{L^2}$ is bounded. What I did not yet manage to show is that the weak derivatives $\partial_{x_i}f$ exist for $i=1,\dots,n$.
Do they even exist? And if so, is there a constant $C$ such that $\|f\|_{C^{0,1}} \le C\|f\|_{W^{1,2}}$ or $\|f\|_{W^{1,2}}\le C\|f\|_{C^{0,1}}$.
I'd be glad for any help or hints to literature on this. 
Thank you very much!

Take a look in the page 279 of this book: "Evans - Partial Differential Equation".
