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## Sobolev Embedding theorem [closed]

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.

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math.stackexchange.com is better suited for this question I think. For the question, you should realize that $W^1_2$ functions are actually equivalence classes of functions. So the statement is that under this equivalence relation there is a continuous version (which is obviously the case for your example). – Thomas Rot Jun 12 2012 at 11:29