Results 1 to 2 of 2

Math Help - riemann integrable

  1. #1
    Member
    Joined
    Jan 2008
    Posts
    114

    riemann integrable

    Let [0,1] ---> R be bounded. Let f be decreasing. By consideration of the sequence of dissection (Dn) of [0,1] show that f is integrable.



    Let E > 0. Then for all N such that s(Dw) < E.

    Hence it satisfies Riemann's criteria for integrability.


    Can some please help me with this question, as my answer as it stands is rubbish and obviously too short, but I don't see what else to say

    Thanks in advance!
    Follow Math Help Forum on Facebook and Google+

  2. #2
    Global Moderator

    Joined
    Nov 2005
    From
    New York City
    Posts
    10,616
    Thanks
    9
    Quote Originally Posted by hunkydory19 View Post
    Let [0,1] ---> R be bounded. Let f be decreasing. By consideration of the sequence of dissection (Dn) of [0,1] show that f is integrable.



    Let E > 0. Then for all N such that s(Dw) < E.

    Hence it satisfies Riemann's criteria for integrability.


    Can some please help me with this question, as my answer as it stands is rubbish and obviously too short, but I don't see what else to say

    Thanks in advance!
    Suppose that f is increasing on [0,1] then if 0\leq x\leq 1 we have f(0)\leq f(x)\leq f(1) thus f is a bounded function. Let P be a partition of [0,1] then difference between the upper and lower sum is \sum_{k=1}^n(\sup\{ f:[x_{k-1},x_k]\}  - \inf \{ f: [x_{k-1},x_k]\} )(x_k-x_{k-1}) = \sum_{k=1}^n (f(x_k) - f(x_{k-1})(x_k - x_{k-1}) \leq \sum_{k=1}^n (f(x_k) - f(x_{k-1})\delta = (f(1) - f(0))\delta where \delta = \max_{1\leq k\leq n} (x_k - x_{k-1}). Thus, if we choose \epsilon < \frac{\delta}{f(1)-f(0)} then we insure that the upper and lower sums are withing \epsilon of eachother. And so the function is integrable.
    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: 7
    Last Post: May 8th 2010, 10:30 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