Results 1 to 2 of 2

Math Help - help with Riemann integrals

  1. #1
    Junior Member
    Joined
    Sep 2009
    Posts
    53

    help with Riemann integrals

    Prove that the function f:[0,1]--> R defined by

    f(x) = {1, if x=1/n, n is an element of Z+
    {0, otherwise

    is Riemann integrable on [0,1].

    Thanks!
    Follow Math Help Forum on Facebook and Google+

  2. #2
    Banned
    Joined
    Oct 2009
    Posts
    4,261
    Thanks
    2
    Quote Originally Posted by friday616 View Post
    Prove that the function f:[0,1]--> R defined by

    f(x) = {1, if x=1/n, n is an element of Z+
    {0, otherwise

    is Riemann integrable on [0,1].

    Thanks!

    1) Using Lebesgue's Theorem: f is Riemann integrable in [0,1] because it is bounded there and the set of points of discontinuity is of measure zero

    2) Directly from the definition: let \sum\limits_{||\Delta||\rightarrow 0}f(c_1)(x_i-x_{i-1}) be the Riemann sum of f with some partition \Delta, and we take the sum when the number of points of the partitions tend to infinity AND the parameter of the partition, i.e. ||\Delta|| goes to zero

    Now, the above sum is zero UNLESS the some of the points c_i happen to be \frac{1}{n} for some n\in \mathbb{N}, and in that case we'll get \sum\limits_{||\Delta||\rightarrow 0}f(c_1)(x_i-x_{i-1})=\sum\limits_{\substack{||\Delta||\rightarrow 0\\c_i=\frac{1}{n}}}(x_i-x_{i-1})

    As the last sum is at most a countable sum and as ||\Delta||\rightarrow 0, for any \epsilon>0 we can choose a partition s.t. x_1-x_{i-1}<\frac{\epsilon}{2^n} , and then \sum\limits_{\substack{||\Delta||\rightarrow 0\\c_i=\frac{1}{n}}}(x_i-x_{i-1})\leq \sum\limits_{n=1}^\infty\frac{\epsilon}{2^n}=\epsi  lon

    Tonio
    Follow Math Help Forum on Facebook and Google+

Similar Math Help Forum Discussions

  1. Evaluating Riemann integrals
    Posted in the Differential Geometry Forum
    Replies: 1
    Last Post: April 18th 2010, 08:31 AM
  2. Riemann integrals
    Posted in the Differential Geometry Forum
    Replies: 1
    Last Post: November 7th 2009, 10:19 PM
  3. error of riemann integrals!!!
    Posted in the Calculus Forum
    Replies: 0
    Last Post: December 1st 2008, 05:06 PM
  4. integrals using Riemann sum
    Posted in the Calculus Forum
    Replies: 1
    Last Post: November 24th 2008, 11:02 PM
  5. Riemann Sums and Riemann Integrals
    Posted in the Calculus Forum
    Replies: 1
    Last Post: October 1st 2006, 01:08 PM

Search Tags


/mathhelpforum @mathhelpforum