Critical values

Critical value may refer to:

Differential topology

In differential topology, a critical value of a differentiable function ƒ : MN between differentiable manifolds is the image (value) ƒ(x) in N of a critical point x in M.[1]

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.


In statistics, a critical value is the value corresponding to a given significance level. This cutoff value determines the boundary between those samples resulting in a test statistic that leads to rejecting the null hypothesis and those that lead to a decision not to reject the null hypothesis. If the calculated value from the statistical test is less than the critical value, then you fail to reject the null hypothesis. If the calculated statistic is outside of the critical value, then you reject the null hypothesis and are forced to accept the alternate hypothesis.

Complex dynamics

In complex dynamics, a critical value is the image of a critical point.


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