What is H°1 (0,1) is it ? Are you talking about Sobolev spaces?
Edit: Assuming this is what you meant, the strongest I could find is that if and then
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.