Let $\Omega\subset\mathbb{R}^N$ be a bounded regular domain and consider the Sobolec space $W^{1,p}(\Omega)$ for $p\in (1,\infty)$. I have some doubts with th trace theorem.
Roughly speaking, the Trace theorem states that there is a linear functional $$T:W^{1,p}(\Omega)\to L^p(\partial \Omega)$$
such that $Tu=u_{|\partial\Omega}$ for every $u\in C^\infty(\overline{\Omega})$.
My questions is:
The only functions for which $Tu=u_{|\partial\Omega}$ are the $C^\infty(\overline{\Omega})$ functions, or can we find, for instance, functions not continuous in $W^{1,p}(\Omega)$ such that the same equality holds?
Thank you