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 lead to a decision not to reject the null hypothesis. If the absolute value of the calculated value from the statistical test is greater than the critical value, then the null hypothesis is rejected and the alternative hypothesis is accepted, and vice versa.