Let $\Sigma$ be an oriented, simply connected surface in $\mathbb{R}^{3}$, let $\xi$ be its unit normal, and let $\alpha$ be a a smooth curve in $\Sigma$.
In this paper, the authors prove that, if every closed curve $\alpha$ has vanishing total torsion, i.e., if $$ \int_{\alpha} \tau = 0 \quad \text{for all closed curves $\alpha$}, $$ then $\Sigma$ is totally umbilic. (In fact, they prove a stronger statement by considering a Riemannian ambient space.)
I am having a hard time understanding a step in their proof of Lemma 2, which establishes the said result.
After proving that the integral of the geodesic torsion $\tau_{g}$ over any closed curve $\alpha$ vanishes, they claim that continuity forces the geodesic torsion to vanish identically; see text after equation (26) in the picture below.
I am not sure why continuity is invoked here. I agree that, if $\nabla_{X}\xi$ and $X$ are not parallel at $p$, then they remain not parallel in a neighborhood of $p$, but how can one then conclude that every closed curve around $p$ has $\tau_{g} =0$?
