  For a complete noncompact connected Riemannian manifold with bounded geometry
$M^n$, we prove that the isoperimetric profile function $I_{M^n}$ is a locally
$\left(1-\frac{1}{n}\right)$-H\"older continuous function and so in particular
it is continuous. Here for bounded geometry we mean that $M$ have $Ricci$
curvature bounded below and volume of balls of radius $1$, uniformly bounded
below with respect to its centers. We prove also the equivalence of the weak
and strong formulation of the isoperimetric profile function in complete
Riemannian manifolds which is based on a lemma having its own interest about
the approximation of finite perimeter sets with finite volume by open bounded
with smooth boundary ones of the same volume. Finally the upper semicontinuity
of the isoperimetric profile for every metric (not necessarily complete) is
shown.
