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Function bounded on [a,b] with finite discontinuities is Riemann integrable

  1. Mar 14, 2012 #1
    1. The problem statement, all variables and given/known data

    to prove that a function bounded on [a,b] with finite discontinuities is Riemann integrable on [a,b]

    2. Relevant equations

    if f is R-integrable on [a,b], then [itex]\forall[/itex] [itex]\epsilon[/itex] > 0 [itex]\exists[/itex] a partition P of [a,b] such that U(P,f)-L(P,f)<[itex]\epsilon[/itex]


    3. The attempt at a solution
    the term on the LHS must be made <ε
    00005wp6.jpg
     
    Last edited: Mar 14, 2012
  2. jcsd
  3. Mar 14, 2012 #2
    You're almost there! You can choose your c_j'' and c_j' to be as close as you want, can't you?
     
  4. Mar 14, 2012 #3
    make it small enough to neglect the second term on the RHS?
     
  5. Mar 14, 2012 #4
    Thats the idea. You can choose c_j''-c_j' to be less than some multiple of epsilon and then proceed.
     
  6. Mar 14, 2012 #5
    got it thanks
    00006qsq.jpg
     
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