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In this note we investigate the behavior of harmonic functions at singularpoints of $\mathsf{RCD}(K,N)$ spaces. In particular we show that their gradientvanishes at all points where the tangent cone is isometric to a cone over ametric measure space with non-maximal diameter. The same analysis is performedfor functions with a Laplacian in $L^{N+\varepsilon}$. As a consequence we showthat on smooth manifolds there is no a priori estimate on the modulus ofcontinuity of the gradient of harmonic functions which depends only on lowerbounds of the sectional curvature. In the same way we show that there is no apriori Calder\'on-Zygmund theory for the Laplacian with bounds depending onlyon lower bounds of the sectional curvature.

The behavior of harmonic functions at singular points of $\mathsf{RCD}$ spaces