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Difficult continuity question

  1. Aug 15, 2007 #1
    1. The problem statement, all variables and given/known data

    We have a worksheet with practice final questions and I'm really stuck on this one on continuity:

    Suppose h: (0,1) -> R has the property that for all x in (0,1), there exists a delta>0 such that for all y in (x, x+delta)[tex]\bigcap[/tex](0,1), h(x) <= h(y)

    a) prove that if h is continuous on (0,1), then h is increasing.
    b) Give a counterexample to show that this need not be true if h is not continuous.

    Thanks so much for any help you can provide!!

    2. Relevant equations



    3. The attempt at a solution



    3. The attempt at a solution
     
  2. jcsd
  3. Aug 15, 2007 #2

    Dick

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    Think about h(x)=1 for x in (0,1/2] and h(x)=0 in (1/2,1). Open boundaries make all the difference.
     
  4. Aug 16, 2007 #3

    HallsofIvy

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    Dick's response is to part (b).

    For (a), Suppose u< v in (0, 1). If h(u)> h(v), can you get a contradiction to "there exists a delta>0 such that for all y in (x, x+delta)(0,1), h(x) <= h(y)" using the intermediate value property?
     
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