I was asked this question in a private message. I am starting a public thread on it because (a) it is an interesting problem, and (b) I am not 100% sure that my answer is correct. So I welcome any criticisms or supplements to my answer.
Here is my answer. Technically, this is a question about the definitions of "critical points" and "local extrema" or perhaps the common understanding among mathematicians of what those terms mean. So the "correct" answer depends on what definitions are applied.Hey, Jeff. Didn't want to create a new thread. This seems like a fairly simple question.
I just wanted to be sure, is this a trick question? I think there are no minima, maxima or critical values in this case, but maybe I don't know something.
If we define "critical point of a function" to mean a point where the first derivative of that function is either zero or undefined, then every real number is a critical point of the greatest integer function because at every real number that is not an integer, the derivative exists and equals zero, and at every integer, the derivative is undefined.
I think a logically sound argument can be made that there is both a local maximum and a local minimum of the greatest integer function at every integer, but that argument would require a very careful definition of what is meant by "local maximum" and "local minimum." I do not know whether that definition is generally acceptable to professional mathematicians.