We consider pointwise linear elliptic equations of the form L_xu_x=η_x on a smooth compact manifold where the operators L_x are in divergence form with real, bounded, measurable coefficients that vary in the space variable x. We establish L²-continuity of the solution at x whenever the coefficients of L_x are L^∞-continuous at x and the initial datum is L²-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 g<ub>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 M with a C¹ heat kernel on a "non-singular" nonempty open subset N, we show that x x↦g_t(x) is continuous whenever x ∈ N.