$M$ is a compact Riemannian manifold and $f$ is a smooth function with the property that for each critical point there is a neighborhood in which $f$ is subharmonic. Can we say that $f$ is subharmonic on $M$?
I am thinking in a way that as $M$ is compact we can cover it by finitely many open sets. we can take there closures as they are also closed set so compact also. So can we use something like this.