Suppose we are considering $I = (0,1)$ and the Sobolev space $W^{1,p}(I)$. Now here is what I know:
If $p > 1$, then $W^{1,p}(I)$ is compactly embedded in $C[0,1]$ (with $||\cdot||_{L^{\infty}}$). This also implies that every $f \in W^{1,p}(0,1)$ is continuous on $C[0,1]$ right (as opposed to merely continuous on $(0,1)$)?
To be more precise on 1., do we know that every $f \in W^{1,p}(0,1)$ is continuous on $[0,1]$, or equal a.e. to a function $\tilde{f}$ continuous on $[0,1]$? As $L^{p}$ functions obviously a.e. functions are equivalent, but if I want to prove a pointwise property of a function in $W^{1,p}(I)$ can I take it to be continuous on $[0,1]$? For example, a problem in Brezis's functional analysis (Exercise 8.8) asks us to show that if for $u \in W^{1,p}(0,1)$ ($p > 1)$ we have $u(x)/x \in L^{p}$, then $u(0) = 0$. Now, if we only knew that $u = \tilde{u}$ a.e. for some $\tilde{u} \in C[0,1]$ then proving the proposition for $\tilde{u}$ would not be enough since it is possible that $u(0) \neq \tilde{u}(0)$. Tldr: Is every function in $W^{1,p}(0,1)$ continuous on $[0,1]$?
What about the case $p = 1$? I know that $W^{1,1}(0,1)$ is not compactly embedded into $C[0,1]$, but is every $f \in W^{1,1}(0,1)$ still in $C[0,1]$?