  We characterize embedded $\C^1$ hypersurfaces of $\R^n$ as the only locally
closed sets with continuously varying flat tangent cones whose
measure-theoretic-multiplicity is at most $m<3/2$. It follows then that any
(topological) hypersurface which has flat tangent cones and is supported
everywhere by balls of uniform radius is $\C^1$. In the real analytic case the
same conclusion holds under the weakened hypothesis that each tangent cone be a
hypersurface. In particular, any convex real analytic hypersurface
$X\subset\R^n$ is $\C^1$. Furthermore, if $X$ is real algebraic, strictly
convex, and unbounded then its projective closure is a $\C^1$ hypersurface as
well, which shows that $X$ is the graph of a function defined over an entire
hyperplane.
