Let $\alpha>0$ and $1\leq p \leq \infty$.
We define a weighted Sobolev space $$X^{\alpha,p}(0,1):= \{u\in W_{loc}^{1,p}(0,1):u\in L^p(0,1), x^\alpha u'\in L^p(0,1)\},$$ where the notation $u\in W_{loc}^{1,p}(0,1)$ means that $u\in W^{1,p}(V)$ for all compact subsets $V$ of $(0,1)$.
I know $u$ is absolutely continuous on $V$ for all compact subsets $V$ of $(0,1).$ But I am not sure how to establish the continuity of $u$ at the right endpoint $x=1$.
In the case that $0<\alpha<1/2,$ $u$ is continuous at $x=0$ as well, for which I do not know how to prove as well.
My question is how to establish the continuity of $u$ at $x=1$ (and at $x=0$ when $0<\alpha<1/2$).
All help will be appreciated! Thanks.