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Product of integrable functions

  1. Jun 22, 2005 #1
    hello all

    im in the middle of proving that if f and g are integrable functions then show that fg is also integrable

    im up to trying to show that M_i(fg,P)-m_i(fg,P) is less than or equal to something that involves U(f,P)-L(f,P)<e^0.5 and U(g,P)-L(g,P)<e^0.5
    anybody have any ideas, if i make any improvements I will post it up

  2. jcsd
  3. Jun 22, 2005 #2


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    Notice that



    [tex]\int_a^b fg dx = \int_a^b \frac{(f+g)^2-(f-g)^2}{4} dx[/itex]

    if that second integral exists. Show that it does.
  4. Jun 23, 2005 #3
    hello there

    well I have spent some time on it but, i cant show that the integral exist because i dont actually know what these functions are, I tried using it with the upper and lower sums but i aint getting anywhere that way

    please help

    thank you
  5. Jun 23, 2005 #4


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    can you, do,it if f,g are positive?
  6. Jun 24, 2005 #5


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    Have you seen the theorem that say that if f and g and integrable, then af+bg (where a,b are constants) is integrable?

    With that and the theorem that (basically) says that if F is integrable and G is continuous, then the composition G(F(x)) is integrable, you show that (f+g)² and (f-g)² are integrable (because x² is continuous and (f+g)² is the composition of f+g by x²)
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