Results 1 to 3 of 3

Math Help - proving the failure of riemann integrability

  1. #1
    Member
    Joined
    Apr 2008
    From
    Seoul, South Korea
    Posts
    128

    proving the failure of riemann integrability

    i want to prove that the function f:[0,1] -> R where [0,1] is the closed interval 0,1 and R is the set of reals, defined by f(x)=1 if x is rational and 0 otherwise, is not riemann integrable. i know that this is the dirischlet's function and that it is not integrable, but i'm not sure how to go about proving it. thanks for any help!
    Follow Math Help Forum on Facebook and Google+

  2. #2
    Global Moderator

    Joined
    Nov 2005
    From
    New York City
    Posts
    10,616
    Thanks
    10
    Because any upper Darboux sum is 1 while any lower Darboux sum is 0.
    This means the upper integral is 1 while the lower integral is 0.
    Thus the function is not Riemann integrable.
    Follow Math Help Forum on Facebook and Google+

  3. #3
    Behold, the power of SARDINES!
    TheEmptySet's Avatar
    Joined
    Feb 2008
    From
    Yuma, AZ, USA
    Posts
    3,764
    Thanks
    78
    Quote Originally Posted by squarerootof2 View Post
    i want to prove that the function f:[0,1] -> R where [0,1] is the closed interval 0,1 and R is the set of reals, defined by f(x)=1 if x is rational and 0 otherwise, is not riemann integrable. i know that this is the dirischlet's function and that it is not integrable, but i'm not sure how to go about proving it. thanks for any help!
    Let P be any sequnce of partitions of [0,1] thats mesh (the length of each sub interval) goes to zero. The upper sum for the integral U(f,p) = 1 by the denisty of the rationals and the lower sum L(f,p)=0 by the density of irrationals

    |U(f,p)-L(f,p)|=1

    so the integral does not exist
    Follow Math Help Forum on Facebook and Google+

Similar Math Help Forum Discussions

  1. [SOLVED] Riemann Integrability of f(x)
    Posted in the Differential Geometry Forum
    Replies: 4
    Last Post: April 4th 2011, 10:28 PM
  2. Riemann Integrability of 1/f
    Posted in the Differential Geometry Forum
    Replies: 3
    Last Post: June 20th 2010, 10:54 AM
  3. Replies: 1
    Last Post: January 16th 2010, 03:31 AM
  4. Riemann integrability
    Posted in the Differential Geometry Forum
    Replies: 5
    Last Post: December 7th 2009, 12:12 PM
  5. Riemann Integrability
    Posted in the Calculus Forum
    Replies: 2
    Last Post: May 15th 2008, 04:22 PM

Search Tags


/mathhelpforum @mathhelpforum