Is it possible that a fonction u(x) element of C[0,1], u(0)=u(1)=0 is not element of H°1 (0,1) , the closure of the H1 Hilbert space ?
I believe it is not possible, but I cannot manage to justify my answer.
Even if and not , we can have the same conclusions ?
The exact question I have is that if and , then is it possible that ?
We never have the hypothesis that and . So, must I conclude that without that stronger hypothesis, we can have a function that is and not ?
Tough one... I don't really know since working with functions in Sobolev spaces is messy as it is, but maybe trying to characterize these functions in easier terms is the best approach. For example: Is a function that is nowhere differentiable weakly differentiable? If the answer is no, then you have the desired function.