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We consider pointwise linear elliptic equations of the form Lxux = ηx on a smooth compact manifold where the operators Lx are in divergence form with real, bounded, measurable coefficients that vary in the space variable x. We establish L 2 -continuity of the solutions at x whenever the coefficients of Lx are L ∞ -continuous at x and the initial datum is L 2 -continuous at x. This is obtained by reducing the continuity of solutions to a homogeneous Kato square root problem. As an application,doi:10.5565/publmat_61117_09 fatcat:7dloi3yitrf27j2wpu4dkukkli