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Help with a proof on integrablitiy

  1. Dec 2, 2008 #1
    1. The problem statement, all variables and given/known data
    Let f:[a,b] [tex]\rightarrow[/tex]R be increasing on the set [a,b] (i.e., f(x)[tex]\leq[/tex] f(y) whenever x<y. Show that f is integrable on [a,b]

    2. Relevant equations
    the definition of integrable that we are using is that [tex]\int[/tex] f=U(f)=L(f)

    3. The attempt at a solution
    What i tried was to start with the fact that we are on an increasing set, which is also compact. I thought since it is compact we know that we are bounded. but then i didn't know how to relate the fact that we are bounded. My thought was that since we have to use U(f) and L(f) which relate to the sup and inf, these must exist since we are on a compact set. I got here and then didn't know where to get that U(f)=L(f) So then i thought i was going in the completely wrong direction, and didn't know where else to go.
  2. jcsd
  3. Dec 2, 2008 #2
    I believe that there is an easy way to get around this by using certain facts about functions with countable number of discontinuties and compact intervals.
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