Hi everyone (1) Let $u\in W^{1,\infty}(0,\infty)$ such that $u(x)>0$ for almost all $x\in(0,\infty)$ and let $\bar u$ its continuous representative. Is It true that $\bar u(x)>0$ for all $x\in [0,\infty)$.
(2) Is It true that : $v\in W^{1,p}(I)$ if and only if $\vert v\vert^p \in W^{1,1}(I)$ ?
(3) Is It true to write that $W^{1,\infty}(]0,\infty[) \hookrightarrow C([0,\infty[)$ et $W^{1,1}(]0,\infty[) \hookrightarrow C([0,\infty[)$ ?
Thanks
(1) You should be able to adapt the function $|x|$ to your needs. This function is positive almost everywhere, but not everywhere.
(2) One implication is standard, and the other is false. For a typical fixed nonnegative function $w \in W^{1,1}$, think of all functions $v$ such that $|v|^p = w$. Most of them are quite wild.
(3) It's not clear to me what you're asking.