Results 1 to 7 of 7

Math Help - Show that f is Riemann Integrable

  1. #1
    Newbie
    Joined
    Sep 2009
    Posts
    23

    Show that f is Riemann Integrable

    f(x)= 0 if x=0,1/2,1/3,1/4,..
    1 otherwise

    So the upper sum is always 1, and I want to show that the upper sum - lower sum is less than epsilon. But how do I proceed for the lower sum?
    Follow Math Help Forum on Facebook and Google+

  2. #2
    Junior Member
    Joined
    Mar 2009
    Posts
    38
    If you have baby Rudin (Principles of Math. Analysis), see Theorem 6.6. The idea I'm thinking of is to take a partition that includes the points you listed except near zero where you can control the width of the "last" segment in the domain. Then add points near the finite number of points from your list. Then you can control the width on which the inf is zero. On the remaining partitions (most of the area) the inf. is 1. So the lower sum can get arbitrarily close to 1.
    Follow Math Help Forum on Facebook and Google+

  3. #3
    MHF Contributor Also sprach Zarathustra's Avatar
    Joined
    Dec 2009
    From
    Russia
    Posts
    1,506
    Thanks
    1
    Quote Originally Posted by mtlchris View Post
    f(x)= 0 if x=0,1/2,1/3,1/4,..
    1 otherwise

    So the upper sum is always 1, and I want to show that the upper sum - lower sum is less than epsilon. But how do I proceed for the lower sum?

    I think you can use the next theorem: If function have Countable set of classification of discontinuities points, then the function is Reimann integrable.

    In your case let \mathbb{M}:=\{0,\frac{1}{n} | n \in \mathbb{N}\} \subset \mathbb{Q} and \mathbb{Q} is Countable set, hence \mathbb{M}  is Countable set.
    .
    .
    .

    And maybe the above is nonsense...
    Follow Math Help Forum on Facebook and Google+

  4. #4
    Newbie
    Joined
    Sep 2009
    Posts
    23
    My book says explicitly to use the theorem that f is in R[a,b] <=> U(p,f)-L(p,f) < epsilon.
    Follow Math Help Forum on Facebook and Google+

  5. #5
    Senior Member roninpro's Avatar
    Joined
    Nov 2009
    Posts
    485
    My first thoughts were similar to huram2215. Did you try his outline?
    Follow Math Help Forum on Facebook and Google+

  6. #6
    Newbie
    Joined
    Sep 2009
    Posts
    23
    I was about to ask for some calrification on this. Like how to write it out?
    Follow Math Help Forum on Facebook and Google+

  7. #7
    Junior Member
    Joined
    Mar 2009
    Posts
    38
    It's important that you get this one on your own. You're up to it ... I can tell from the initial post. Use the technique I outlined; it gives you a key idea used in many problems in real analysis (Lebesgue integration) later. So it's important to your development.

    To help you start, choose some \epsilon > 0. Then choose N so that \frac{1}{N} < \frac{\epsilon}{2}. Now you have a finite number of points outside the segment [0, \frac{1}{N}] and still have \frac{\epsilon}{2} to spend. Choose intervals that contain the remaining (finite number of) points in a way that serves your purpose.
    Follow Math Help Forum on Facebook and Google+

Similar Math Help Forum Discussions

  1. f & g Riemann integrable, show fg is integrable
    Posted in the Differential Geometry Forum
    Replies: 4
    Last Post: February 12th 2011, 09:19 PM
  2. Riemann integrable
    Posted in the Differential Geometry Forum
    Replies: 3
    Last Post: March 18th 2010, 04:25 AM
  3. Riemann Integrable
    Posted in the Calculus Forum
    Replies: 0
    Last Post: November 7th 2009, 01:20 PM
  4. riemann integrable
    Posted in the Calculus Forum
    Replies: 1
    Last Post: April 29th 2009, 05:48 AM
  5. riemann integrable
    Posted in the Calculus Forum
    Replies: 2
    Last Post: December 4th 2008, 07:29 AM

Search Tags


/mathhelpforum @mathhelpforum