Results 1 to 3 of 3

Math Help - To prove a function is integrable (or not)

  1. #1
    Super Member
    Joined
    Mar 2006
    Posts
    705
    Thanks
    2

    To prove a function is integrable (or not)

    Let g:[0,2]\mapsto\mathbb{R} be defined by g(x)=\left\{\begin{array}{cc}x,&\mbox{ if }x\epsilon\mathbb{Q}\\-x, & \mbox{ if } x\epsilon\mathbb{R}/\mathbb{Q}\end{array}\right.

    (a) Let P = {0,1,2}. Evaluate U(P,g).
    (b) Let P = {0,1,2}. Evaluate L(P,g).
    (c) Define a partition Q by Q={x_0,x_1,...x_n} where 0=x_0<x_1<...<x_{n-1}<x_n=2. Prove that U(Q,g) \geq 2.
    (d) Is the function g integrable on the interval [0,2]? If so, evaluate \int^{2}_{0}g(x)dx. If not, explain why it is not integrable on [0,2].

    My works so far:

    (a) 3
    (b) -3
    (c) ....still working
    (d) ....still working

    Sorry about the amount of problems that I have posted up lately, it is just that there are a lot of new materials that I don't understand from recent lectures (we are working on integrations now), and it is just impossible for me to go to the professor for every single problem.

    Thank you very much!

    KK
    Follow Math Help Forum on Facebook and Google+

  2. #2
    Global Moderator

    Joined
    Nov 2005
    From
    New York City
    Posts
    10,616
    Thanks
    10
    The theorem says that a function defined on a closed interval is Riemann integratble if and only if it is almost eveywhere continous. But maybe you did not do that theorem and are asked to show that a bounded ,defined almost discontinous function is not Riemann integrable. If thus, here is something that might help http://www.mathhelpforum.com/math-he...stion-4-a.html. The trick here was to subdivide the the region into rational and then irrational points.
    Follow Math Help Forum on Facebook and Google+

  3. #3
    Super Member
    Joined
    Mar 2006
    Posts
    705
    Thanks
    2
    So would it be proper to say that this function is not integrable since U(P,g) doesn't equal to L(P,g) for any partition P in g?
    Follow Math Help Forum on Facebook and Google+

Similar Math Help Forum Discussions

  1. [SOLVED] Prove that a function is Riemann integrable?
    Posted in the Differential Geometry Forum
    Replies: 4
    Last Post: August 29th 2010, 11:57 AM
  2. Prove that this function is integrable
    Posted in the Differential Geometry Forum
    Replies: 2
    Last Post: December 22nd 2009, 11:06 AM
  3. Prove that a function is integrable
    Posted in the Differential Geometry Forum
    Replies: 1
    Last Post: November 29th 2009, 07:32 PM
  4. Replies: 0
    Last Post: December 1st 2008, 07:43 PM
  5. Prove a function is integrable
    Posted in the Calculus Forum
    Replies: 12
    Last Post: February 12th 2007, 10:51 AM

Search Tags


/mathhelpforum @mathhelpforum