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Homework Help: Intermediate Value Property

  1. Jun 27, 2016 #1
    1. The problem statement, all variables and given/known data
    upload_2016-6-27_9-0-49.png '
    Here is the given problem

    2. Relevant equations

    3. The attempt at a solution

    a. For part a, I felt it was not continuous because of the sin(1/x) as it gets closer to 0, the graph switches between 1 and -1. Then I felt it might be continuous, therefore I am not sure.

    b. For part b, I felt it has the Intermediate Value Property (IVP), because I can do something with the IVT. Those were my thoughts and ideas.
    Last edited by a moderator: Jun 27, 2016
  2. jcsd
  3. Jun 27, 2016 #2


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    Can you show that the limit as approached positively is different from that as approached negatively? Or that it's different than f(0)? So in other words,
    ##\lim_{x \to 0} f(x) \neq f(0)##?

    For b, I'm not really sure. I thought one of the requirements was that f(x) was continuous...
    Perhaps this will be of some use.
    Looks like you need to look at the derivatives near zero.
  4. Jun 28, 2016 #3
    For part a you would have to do what BiGyElLoWhAt suggested. For b I believe you would have to prove that the function in either monotone increasing or decreasing. IVP says that for any x value between two other x values, the y value will be in between the y values for the other two x values.
    Last edited: Jun 28, 2016
  5. Jun 28, 2016 #4
    IVP theorem
    Last edited: Jun 28, 2016
  6. Jun 28, 2016 #5
  7. Jul 2, 2016 #6


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    If there exist sequences [itex]x_n[/itex] and [itex]y_n[/itex] such that [itex]\lim_{n \to \infty} x_n = \lim_{n \to \infty} y_n = 0[/itex] but [itex]\lim_{n \to \infty} f(x_n) \neq \lim_{n \to \infty} f(y_n)[/itex] then [itex]f[/itex] is not continuous at zero.
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