We are trying to show that the map $f\mapsto|f|$ is a continuous (nonlinear) map from $W^{1,p}(\Omega)\to W^{1,p}(\Omega)$ for any bounded/open region $\Omega$ and for $p\in[1,\infty)$.
We have tried using the regular definition of continuity using the standard norm on Sobolev spaces for both $f$ and $|f|$:
$\displaystyle\|u\|^p=\sum_{\alpha:|\alpha|\leq1}\|\partial^\alpha u\|^p_{L^p(\Omega)}$
where $\alpha$ is a multi-index and $|\alpha|$ denotes the sum of its components, but we unable to make any progress. Is there another approach we should try, or some trick that we seem to be forgetting? Thank you for your time.