# Regular value theorem

Let $M, N$ be differential manifolds and $p \in N$ be a regular value of a differentiable map $f: M \to N$. Then $f^{-1}(p)$ is a submanifold of $M$.
A slightly stronger version of this result states the following: if there is an open neighbourhood $U$ of $p$ in $N$ such that the rank of the Jacobian is constant for all points in $f^{-1}(U)$, then $f^{-1}(p)$ is a submanifold of $M$.