For a system of ODEs defined on an open, convex domain $U$ containing a
positively invariant set $\Gamma$, we prove that under appropriate hypotheses,
$\Gamma$ is the graph of a $C^r$ function and thus a $C^r$ manifold. Because
the hypotheses can be easily verified by inspecting the vector field of the
system, this invariant manifold theory can be used to study the existence of
invariant manifolds in systems involving a wide range of parameters and the
persistence of invariant manifolds whose normal hyperbolicity vanishes when a
small parameter goes to zero. We apply this invariant manifold theory to study
three examples and in each case obtain results that are not attainable by
classical normally hyperbolic invariant manifold theory.