Suppose that we have a map $f: M \to N$ between two smooth manifolds, which is only $C^{1}$, that is differentiable with continuous differential. Let $n \in N$ be a regular value of $f$, that is, $d_mf$ is surjective, at any $m$ such that $f(m)=n$.
What is the class of the regular subspace $f^{-1}(n)$ ? Is it only $C^{1}$ ?
Thanks !