I have a problem with Sobolev Embedding theorems. I can't understand how the space $W^1_2(0,1)$ embeds into $C(0,1)$. According to Sobolev's theorems that should be the case. However, as far as I understand, the function $u(x)=x$ when $x\in [0,1]\backslash 0.5$ and $u(x)=10$ when $x=0.5$ belongs to $W_2^1(0,1)$ and should be continuous but it is not.
closed as too localized by Bill Johnson, Deane Yang, Daniel Moskovich, Denis Serre, Michael Renardy Jun 12 2012 at 14:43