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Showing IVP with composition of functions

  1. Sep 21, 2012 #1
    1. The problem statement, all variables and given/known data
    Suppose that f has the intermediate value property on an interval J, that g has the intermediate value property on an interval I and that g(I) is a subset of J. Prove that f°g has intermediate value property on I.


    2. Relevant equations



    3. The attempt at a solution

    I think I might have simplified too much and missed the point. Here is my work so far...


    Since f has IVP on J, there is an a,b in J and a≠b and v is a number between f(a) and f(b) such that there is a c between a and b that f(c)=v.

    Similarly for g, there is a e,f in I and e≠f and u is a number between g(e) and g(f) such that there is a d between e and f that g(d)=u.

    Since g(I) is a subset of J, f(g(I)) is a subset of f(J).

    Then f(g(e)) and f(g(f)) are numbers such that f(g(e))≠f(g(f)). Since g(d)=u is between e and f, then f(u) is between f(g(e) and f(g(f)) since f has IVP on J.

    ∴f°g has IVP on I.
     
  2. jcsd
  3. Sep 21, 2012 #2

    SammyS

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    First of all -- and this is a good place to start -- What do you need to show to prove that f°g has intermediate value property on I ?
     
  4. Sep 22, 2012 #3

    Yes, that is what I am hoping to show.
     
  5. Sep 22, 2012 #4

    SammyS

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    No. What I meant was:

    What is it that needs to be shown in order to prove that f○g has intermediate value property on I ?

    In other words: How does f○g need to behave to demonstrate that f○g has the intermediate value property on the interval, I ?
     
  6. Sep 23, 2012 #5
    Sorry, I misread.

    That for some distinct a,b in f(g(I)), for v between f(g(a)) and f(g(b)) there is a c in I that f(g(c))=v is what needs to be shown I think.
     
  7. Sep 23, 2012 #6

    SammyS

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    For one thing, a & b are in I not in f(g(I)) .

    For another, this must be true for every a,b in I such that a < b .

    So, to go for there, for any a,b in I such that a < b, what does the fact that g has the intermediate value property on I tell you about v between g(a) and g(b) ?
     
  8. Sep 23, 2012 #7
    There is a c in I so that g(c)=v. Then this extends to the composition?
     
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