My question is: how can one prove that if $\phi\in H^{n}(a,b)$ for all positive integer $n$, then $\phi\in C^{\infty}(a,b)$?
$H^{n}(a,b)$ denotes de Sobolev space of the real interval $(a,b)$. For my purposes one can assume that $\phi(a)=\phi(b)=0$.
I understand that there is actually a stronger result: if $\phi\in H^{n+1}(a,b)$, then $\phi\in C^{n}(a,b)$ (something which I know is true for $n=0$). In other words, having $n+1$ weak derivatives, implies that the function has at least $n$ strong derivatives.
Is this true? Can anyone give me a sketch of the proofs?
It is sufficient to prove that: $\phi\in H^1(a,b)$ implies $\phi\in C^0{(a,b)}$, but this follows from Theorem 8.2 of Brezis book.