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Prove increasing function defines everywhere is Rinemann-integrable

  1. Nov 25, 2007 #1

    quasar987

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    1. The problem statement, all variables and given/known data
    This is probably easy but i cant seem to see how to make it work right now.

    I am trying to show that if we have a function f:R-->R that is increasing, then for any interval [a,b], it is riemann-integrable.

    I know it's true because a book I saw refers to another book for the proof that an increasing fct as a countable number of discontinuities.


    3. The attempt at a solution
     
  2. jcsd
  3. Nov 25, 2007 #2
    If you know the proof of the fact that f is Riemann integrable iff it is continuous (lambda) almost everywhere then try to work it in...countable sets have 0 measure, of course.

    Ie, suppose f is discontinuous on some non-null set and derive a contradiction.
     
  4. Nov 25, 2007 #3

    quasar987

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    Is this just an idea you're throwing at me, or do you know for a fact that it works?

    (As a matter of fact, I have proven that very theorem in an earlier homework sheet for this course)
     
  5. Nov 25, 2007 #4

    CompuChip

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    It will work. Try it :smile:
     
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