Differential Topology
In differential topology, a critical value of a differentiable function ƒ : M → N between differentiable manifolds is the image (value) ƒ(x) in N of a critical point x in M.
The basic result on critical values is Sard's lemma. The set of critical values can be quite irregular; but in Morse theory it becomes important to consider real-valued functions on a manifold M, such that the set of critical values is in fact finite. The theory of Morse functions shows that there are many such functions; and that they are even typical, or generic in the sense of Baire category.
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