Critical value

Definition

Let $f:M\to N$ be a smooth map between differential manifolds $M$ and $N$. A point $p\in N$ is termed a critical value for $f$ if it satisfies the following condition:

• There exists $q\in f^{-1}(p)$ such that $q$ is a critical point; in other words, the map $(Df):T_{q}M\to T_{p}N$ is not surjective. In other words, it is the image of a critical point.
• It is not a regular value of $f$.

$N$ is thus partitioned into two subsets: the critical values of $f$, and the regular values of $f$.