Critical value

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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_qM \to T_pN 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.