I'm looking for conditions that ensure the closure of an embedded manifold is a manifold with boundary.

The specific case I'm interested is as follows. Consider an oriented $C^1$ surface ${\cal S}$ in $\Bbb R^3$ such that its closure is known to be a $C^0$ surface with boundary. Moreover, suppose we know that the normal to ${\cal S}$ has a continuous extension to the closure. Does this imply that the closure is a $C^1$ manifold with boundary?