Sobolev embeddings

Question

I'm doing some reading on embeddings of Sobolev spaces and at the moment I am trying to understand why $H^1(0,1)\subset C(0,1)$. The proof I found basically shows that for any $u\in H^1(0,1)$ the inequality $$||u||_\infty \leq c ||u||_{H^1(0,1)}$$ holds for some constant $c>0$, but I don't understand yet why this proves the subset statement.

Answer 1

Fix $x_0\in (0,1)$. The previous inequality gives, by a scaling argument, that $$|u(x)-u(x_0)|\leqslant C\cdot\left(\int_{(x_0-\delta,x_0+\delta)}(|u(t)|^2+|u'(t)|^2)dt\right)^{1/2}$$ if $|x-x_0|<\delta$, hence choosing $\delta$ small enough, we get that $u$ is in the equivalence class for equality almost everywhere of a continuous function.