  Let $M$ be a compact manifold of dimension $n$. In this paper, we introduce
the {\em Mass Function} $a \geq 0 \mapsto \xp{M}{a}$ (resp. $a \geq 0 \mapsto
\xm{M}{a}$) which is defined as the supremum (resp. infimum) of the masses of
all metrics on $M$ whose Yamabe constant is larger than $a$ and which are flat
on a ball of radius~$1$ and centered at a point $p \in M$. Here, the mass of a
metric flat around~$p$ is the constant term in the expansion of the Green
function of the conformal Laplacian at~$p$. We show that these functions are
well defined and have many properties which allow to obtain applications to the
Yamabe invariant (i.e. the supremum of Yamabe constants over the set of all
metrics on $M$).
