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Math Help - Discontinuous Function, Critical Points, Local Extrema

  1. #1
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    Discontinuous Function, Critical Points, Local Extrema

    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.

    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.


    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.

    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.
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    Re: Discontinuous Function, Critical Points, Local Extrema

    WHAT definition? You are spend a lot of time saying you need to have a "careful definition of what is meant by 'local maximum' and 'local minimum'" but you don't say what definitions you want to use. The usual are "a function, f(x), has a 'local maximum' at x= x_0 if and only if there exist a neighborhood of x_0 such that f(x)\le f(x_0) with x in that neighborhood and similarly, with the inequality reversed, for 'local minimum'.
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    Re: Discontinuous Function, Critical Points, Local Extrema

    Quote Originally Posted by HallsofIvy View Post
    WHAT definition? You are spend a lot of time saying you need to have a "careful definition of what is meant by 'local maximum' and 'local minimum'" but you don't say what definitions you want to use. The usual are "a function, f(x), has a 'local maximum' at x= x_0 if and only if there exist a neighborhood of x_0 such that f(x)\le f(x_0) with x in that neighborhood and similarly, with the inequality reversed, for 'local minimum'.
    Using that definition, then there is a local minimum and local maximum of the function at every real x that is not an integer, and every integer is a local maximum.
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