I am having some trouble proving the following. Let $I \subset \mathbb{R}$ and consider $u \in H^1(I)$. Fix some point $y_0 \in I$, and define
$$ \bar{u}(x) := \int_{y_0}^x u'~dt, $$ then $\bar{u}(x)$ is continuous over $\bar{I}$.
This might seem silly but it doesnt seem trivial to me. Any help would be appreciated.
$$|\overline{u}(y)-\overline{u}(x)|\leq\int_x^y |u'(t)|\text{d}t \leq \sqrt{\int_x^y1^2\text{d}t} \sqrt{\int_x^y |u'(t)|^2\text{d}t}=\sqrt{|y-x|} \|u'\|_{L^2(I)}$$