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A property of a riemann stieltjes integral

  1. Jul 29, 2011 #1
    Hi!
    While studying a text " A First Course in Real Analysis" by protter, I've been asked to prove a property of riemann stieltjes integral.
    The propery is as follows ; Suppose a<c<b. Assume that not both f and g are discontinuous at c. If [itex]\int[/itex]fdg from a to c and [itex]\int[/itex]fdg ffrom c to b exist, then
    [itex]\int[/itex]fdg from a to b exists and [itex]\int[/itex]fdg from a to b = [itex]\int[/itex]fdg from a to c +[itex]\int[/itex]fdg from c to b.

    This is written in p.317 of the book.
    What I want to ask you is if this property is correct or not.
    In some books, incorrect theorems are sometimes introduced. So, those things make me to doubt other books, including the above book.
    Thank you for reading my long questions.
     
  2. jcsd
  3. Jul 29, 2011 #2

    HallsofIvy

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    You are asked to prove that, with a< c< b, and both [itex]\int_a^c fdg[/itex] and [itex]\int_c^b fdg[/itex] exists, then [itex]\int_a^b fdg[/itex] exists and
    [tex]\int_a^b fdg= \int_a^c fdg+ \int_c^b fdg[/tex]

    Yes, that is perfectly true and is an important property of an integral. The key point of the proof is that for any partition of [a, b], we can use a refinement that includes the point c.
     
  4. Jul 29, 2011 #3
    Do you mean that the assumption "not both f and g are discontinuous at c" is not needed to prove it??. If not, I want to know where the assumption is used in the proof and where I can find the proof of it.
    Could you tell me about those things??
     
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