subharmonic function and support functions

Question

$M$ is a Riemannian manifold and $f$ is a continuous function on $M$. $f$ has the property that for any $p \in M,\epsilon>0$, we can find a smooth function $f_{\epsilon}$ such that ${f_\varepsilon }(p) = f(p)$, ${f_\varepsilon }(x) \le f(x)$ in a small neighborhood of $p$, and $\Delta {f_\varepsilon }(p) \ge 0$. Can we conclude that $f$ is subharmonic in the sense of distribution, that is, for any non-negative function $\varphi \in C_c^\infty (M)$, we have $\int {f\Delta \varphi } {\rm{d}}V \ge 0$? Moreover, is the converse conclusion true?