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