  A classical theorem in conformal geometry states that on a manifold with
non-positive Yamabe invariant, a smooth metric achieving the invariant must be
Einstein. In this work, we extend it to the singular case and show that in all
dimension, if a continuous metric is smooth outside a compact set of high
co-dimension and achieves the Yamabe invariant, then the metric is Einstein
away from the singularity and can be extended to be smooth on the manifold in a
suitable sense. As an application of the method, we prove a Positive Mass
Theorem for asymptotically flat manifolds with analogous singularities.
