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Functions and the Intermediate Value theorem

  1. Nov 17, 2009 #1
    1. The problem statement, all variables and given/known data
    Let f : [0; 1] --> R be continuous on [0, 1], and assume that the range of f is contained
    in [0; 1]. Prove that there exists an x in [0, 1] satisfying f(x) = x.


    2. Relevant equations



    3. The attempt at a solution

    Well i am almost positive I need to use the intermediate value theorem.

    First I could claim that either f(0)>x>f(1) or f(0)<x<f(1). where x is a value in (0,1)
    Not too sure what to do, i think the key is something to do with the range of f being contained in [0,1], but any help would be great!
     
  2. jcsd
  3. Nov 17, 2009 #2

    HallsofIvy

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    If f(0)= 0, we are done. If f(1)= 1, we are done. So we can assume that [itex]f(0)\ne 0[/itex] and that [itex]f(1)\ne 1[/itex]. But f(0) must be in [0, 1]. If it is not equal to 0, then f(0)> 0. Similarly, we must have f(1)< 1. Define H(x)= f(x)- x. What is H(0)? What is H(1)? Apply the intermediate value theorem to H(x).
     
  4. Nov 17, 2009 #3
    Thanks so much for the help!
     
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