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Intermediate Value Theorem

  1. Oct 12, 2012 #1
    1. The problem statement, all variables and given/known data
    a)Let f(x) be continuous on [0, 2], with f(0) = f(2). Show that f(x) = f(x+1) for some x ε [0, 1].
    b)Let f(x) be 1:1 and continuous on the interval [a, b] with f(a) < f(b). Show that the range of f is the interval [f(a), f(b)].


    2. Relevant equations



    3. The attempt at a solution
    I'm not really where to start for either of them. In a), I find it obvious that there exists an f(x) = f(x+1) for some x in that interval, but find it difficult to prove without any specific function. I find using the I.V.T. difficult in general without being applied to a specific function. Any help/hints appreciated. Thanks!
     
  2. jcsd
  3. Oct 12, 2012 #2
    Hello fellow Waterloo student! I would love to help, but alas, I am having the same problems as you! :S
     
  4. Oct 12, 2012 #3
    Hello busterkomo try taking a function h such that:

    h(x)=f(x)-f(x+1)

    And h is obviously a continuous function as a difference of two continuous functions.
    Now what can you do from that??
     
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