Let $H^0 (\mathbb{R}^N) := L^2 (\mathbb{R}^N)$. Suppose that $v \in \bigcap_{s \in [0,1)} H^s (\mathbb{R}^N) \supseteq H^1 (\mathbb{R}^N)$ and additionaly, $v \in H^1_{loc} (\mathbb{R}^N)$. Can one show that $v \in H^1 (\mathbb{R}^N)$? If not, what is the example of such function?
As far as I can see, it is clear that $v \in L^2 (\mathbb{R}^N)$ and it has a weak derivative $\nabla v \in L^2_{loc} (\mathbb{R}^N)$. Is true that $\nabla v \in L^2 (\mathbb{R}^N)$?