Results 1 to 3 of 3

Math Help - riemann integral

  1. #1
    Newbie
    Joined
    Mar 2008
    Posts
    1

    riemann integral

    If f is continous on [a,b],then f belongs to R[a,b],R-RIEMANN INTEGRAL
    Follow Math Help Forum on Facebook and Google+

  2. #2
    is up to his old tricks again! Jhevon's Avatar
    Joined
    Feb 2007
    From
    New York, USA
    Posts
    11,663
    Thanks
    3
    Quote Originally Posted by yarlagaddatrinath View Post
    If f is continous on [a,b],then f belongs to R[a,b],R-RIEMANN INTEGRAL
    this question is phrased weird. what do you mean "belongs to R[a,b]"? do you mean it is Riemann Integrable?
    Follow Math Help Forum on Facebook and Google+

  3. #3
    Global Moderator

    Joined
    Nov 2005
    From
    New York City
    Posts
    10,616
    Thanks
    10
    Quote Originally Posted by yarlagaddatrinath View Post
    If f is continous on [a,b],then f belongs to R[a,b],R-RIEMANN INTEGRAL
    A continous function on [a,b] is always Riemann integrable. Since f is continous it is bounded by extreme value theorem. Furthermore, continous functions on compact (in particular closed) sets are uniformly continous. To prove Riemann integrability we need to show S_P(f) - s_P(f) < \epsilon where P is a partition of [a,b]. Given any \epsilon > 0 there is a \delta > 0 such that |x-y|<\delta \mbox{ and }x,y\in [a,b]\implies |f(x)-f(y)|<\epsilon. Pick a partition P such that  (x_i - x_{i-1}) < \delta. Then f assumes its maximum and minimum on each interval and so S_P(f) - s_P(f) < \epsilon(b-a), and this quantity can be made arbitrary small.
    Follow Math Help Forum on Facebook and Google+

Similar Math Help Forum Discussions

  1. Riemann integral
    Posted in the Calculus Forum
    Replies: 2
    Last Post: November 28th 2011, 08:29 AM
  2. Riemann integral
    Posted in the Calculus Forum
    Replies: 4
    Last Post: September 11th 2010, 03:03 PM
  3. Riemann Sum to an integral
    Posted in the Calculus Forum
    Replies: 2
    Last Post: November 30th 2009, 11:33 AM
  4. Riemann Integral
    Posted in the Differential Geometry Forum
    Replies: 0
    Last Post: August 17th 2009, 02:47 PM
  5. Replies: 6
    Last Post: August 4th 2007, 10:48 AM

Search Tags


/mathhelpforum @mathhelpforum