The following is the trace theorem in Partial Differential Equations by Evans:
Let $U$ be a domain (open connected subset) of $\mathbb{R}^n$. Suppose $U$ is bounded and $\partial U$ is $C^1$. Then there exists a bounded linear operator $$ T:W^{1,p}(U)\to L^p(\partial U) $$ such that $Tu=u\mid_{\partial U}$ if $u\in W^{1,p}(U)\cap C(\overline{U})$ and $$ \|Tu\|_{L^p(\partial U)}\leq C\|u\|_{W^{1,p}(U)} $$ for each $u\in W^{1,p}(U)$, with $C=C(p,U)$.
I don't know how different $W^{1,p}(U)$ and $C(\overline{U})$ can be. Would anybody help me with simple examples in $W^{1,p}(U)\setminus C(\overline{U})$ and $C(\overline{U})\setminus W^{1,p}(U)$?
Continuous but not in any Sobolev space: let $f(x)=g(x_1)$ where $g$ is a continuous but not absolutely continuous: Cantor staircase, or a Weierstrass-type nowhere differentiable function. Since Sobolev functions are absolutely continuous on almost every line segment parallel to a coordinate axis, $f$ is not in $W^{1,p}$.
In $W^{1,p}(U)$ but without a continuous extension to $\overline{U}$: this requires $p\le n$, because for $p>n$ the Morrey-Sobolev inequality gives Hölder continuity, and that propagates to the boundary. Pick a point $x_0\in \partial U$ and let $u(x)=\log \log \frac{1}{\|x-x_0\|}$. This function is in $W^{1,p}(U)$ for $1\le p\le n$ (provided $n\ge 2$), but has no continuous extension to the boundary.