  We consider pointwise linear elliptic equations of the form $\mathrm{L}_x u_x
= \eta_x$ on a smooth compact manifold where the operators $\mathrm{L}_x$ are
in divergence form with real, bounded, measurable coefficients that vary in the
space variable $x$. We establish $\mathrm{L}^{2}$-continuity of the solutions
at $x$ whenever the coefficients of $\mathrm{L}_x$ are
$\mathrm{L}^{\infty}$-continuous at $x$ and the initial datum is
$\mathrm{L}^{2}$-continuous at $x$. This is obtained by reducing the continuity
of solutions to a homogeneous Kato square root problem. As an application, we
consider a time evolving family of metrics $\mathrm{g}_t$ that is tangential to
the Ricci flow almost-everywhere along geodesics when starting with a smooth
initial metric. Under the assumption that our initial metric is a rough metric
on $\mathcal{M}$ with a $\mathrm{C}^{1}$ heat kernel on a "non-singular"
nonempty open subset $\mathcal{N}$, we show that $x \mapsto \mathrm{g}_t(x)$ is
continuous whenever $x \in \mathcal{N}$.
