1. PF Contest - Win "Conquering the Physics GRE" book! Click Here to Enter
    Dismiss Notice
Dismiss Notice
Join Physics Forums Today!
The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

Riemann integration

  1. Feb 18, 2008 #1
    1. The problem statement, all variables and given/known data
    Prove that the function specified below is Riemann integrable and that its integral is equal to zero.

    2. Relevant equations
    f(x)=1 for x=1/n (n is a natural number) and 0 elsewhere on the interval [0,1].

    3. The attempt at a solution
    I have divided the partition into two subintervals, the first with tags different from x=1/n and the second with tags at x=1/n. But, given an epsilon>0, I am not sure how to choose my delta (the norm of the partition) such that the points where the function is not zero doesn't make a contribution.

    Or, is my approach all wrong?

  2. jcsd
  3. Feb 18, 2008 #2


    User Avatar
    Science Advisor
    Homework Helper
    Gold Member

    Maybe not all wrong, but I would say overly complicated. :)

    Consider any partition of [0,1]. Note that every subinterval from your partition contains a point not of the form 1/n.
  4. Feb 18, 2008 #3
    Yeah, it is sometimes like that if you study independently. :P

    So, considering any partition of [0,1]. I should then tag the points different from 1/n, then making all the contributions zero. Right?
  5. Feb 18, 2008 #4


    User Avatar
    Science Advisor
    Homework Helper
    Gold Member

    Wait, a sec, have you seen the result that if a function is discontinuous at a countable number of points then it is integrable?
  6. Feb 18, 2008 #5
    No, i have not. But I will definetly look for it now.

  7. Feb 18, 2008 #6


    User Avatar
    Science Advisor
    Homework Helper
    Gold Member

    Well, if your book hasn't covered this yet, try to do without it.

    You want to show that the lower and upper integrals are 0. Prove that the lower riemann sums s(f;P) are 0 for any partition P of [0,1]. This will of course imply that the lower integral is 0.

    For the upper integral, you want to show that the inf over every P partition of [0,1] of the upper riemann sums S(f,P) is 0. Show that for every epsilon>0, you can always find a partition P' such that S(f;P')<epsilon.
  8. Feb 18, 2008 #7
    What you are describing now feels much better, the squeeze theorem. :)

    But I don't understand at all, how to deal with the upper integral...? When finding the inf over every partition. I would like to do it in the same way as i treat the lower integral.

    Last edited: Feb 18, 2008
  9. Feb 18, 2008 #8


    User Avatar
    Science Advisor
    Homework Helper
    Gold Member

    And I don't understand your question. :P
  10. Feb 18, 2008 #9
    Ok. :)

    I don't understand this!
  11. Feb 18, 2008 #10
    Now, I understand. (I hope so anyway)

    When you wrote 'inf' I thought you meant infimum... so I thought that I was really lost since I have never heard of infimum in the context as Riemann integrals. But you must have meant int as in integral, right?

    And, yes! I am an analysis-rookie. ;)
  12. Feb 18, 2008 #11


    User Avatar
    Science Advisor
    Homework Helper
    Gold Member

    I sure meant infimum.

    How do you define the upper integral? For me, the upper integral is defined as

    [tex]\inf_{P\in \mathcal{P}[0,1]}S(f;P)[/tex]

    where [tex]\mathcal{P}[0,1][/tex] is the set of all partitions of [0,1] and

    [tex]S(f;P)=\sum_{x_i\in P}\max_{x\in[x_i-1,x_i]}f(x)[/tex]
  13. Feb 18, 2008 #12

    I don't define the 'upper integral' at all. For me the 'Riemann integral' is defined as a limit of the Riemann sums as the norm tend to zero. That is why I am talking about the partitions and their tags. I can't find any section with upper Riemann sums either. It is only the Riemann sum.
  14. Feb 18, 2008 #13


    User Avatar
    Science Advisor
    Homework Helper
    Gold Member

    I see!

    Well in that case it's even simpler! Consider epsilon>0, then for any delta>0, we have that any Riemann sum associated with a partition whose norm is lesser than delta is 0 because in every subinterval of [0,1], there is a point not of the form 1/n!
  15. Feb 18, 2008 #14


    User Avatar
    Homework Helper

    I would procede thusly
    clearly the lower integral is 0
    take any partition and choose taggeg point where f=0
    consider the upper integral
    suppose the norm is h where 1>h>0
    The idea is we want to make a large sum
    f=0,1 so we choose f=1 whenever possible
    to take maximum advantage consiger our taggged partition
    we would like n for our tagged partition to include points where f=1 when possible
    so we begin
    however at some point intervals chosen in this way begin to intersect and an adjustment is needed
    we need to know when
    elementary algebra tells us this happens when
    the upper integral
    we want something in h alone
    I leave that to you
  16. Feb 19, 2008 #15
    Finally, I understand! Thank you so much! :)
Know someone interested in this topic? Share this thread via Reddit, Google+, Twitter, or Facebook

Similar Threads - Riemann integration Date
Riemann integrability Dec 18, 2017
Number of subdivisions in a Riemann integral (DFT) Feb 4, 2017
Riemann sum - integral Jan 4, 2017
Definite integral as Riemann sums Nov 8, 2016