Embedded Smooth Hypersurface from Regular Points of Smooth Function

Theorem
Let $\struct {M, g}$ be a Riemannian manifold.

Let $f \in \map {C^\infty} M$ be a smooth function.

Let $R \subseteq M$ be the set of regular points of $f$.

Let $c \in \R$.

Let $M_c$ be a set such that:


 * $M_c = \map {f^{-1} } c \cap R$

Suppose $M_c$ is non-empty.

Then $M_c$ is an embedded smooth hypersurface in $M$.

Furthermore, the gradient $\grad f$ is everywhere normal to $M_c$.