Results 1 to 5 of 5

Math Help - Reimann Integrable

  1. #1
    Member thaopanda's Avatar
    Joined
    Sep 2009
    From
    Worcester, Massachusetts
    Posts
    85

    Reimann Integrable

    Determine whether the function f: [0,1] \rightarrow R given by:

    f(x):=
    3 if x \in Q \cap [0,1]
    1 if x \in (R \  Q) \cap [0,1]

    is Riemann integrable on [0,1].

    I know that if it is Riemann integrable, sup L(P, f) = inf U(P,f), P being the partition of [0,1]
    or \int f(x)dx (lower integral of f over [0,1]) = \int f(x)dx (upper integral of f over [0,1])

    How do I show if it is or not?
    Follow Math Help Forum on Facebook and Google+

  2. #2
    MHF Contributor Drexel28's Avatar
    Joined
    Nov 2009
    From
    Berkeley, California
    Posts
    4,563
    Thanks
    21
    Quote Originally Posted by thaopanda View Post
    Determine whether the function f: [0,1] \rightarrow R given by:

    f(x):=
    3 if x \in Q \cap [0,1]
    1 if x \in (R \  Q) \cap [0,1]

    is Riemann integrable on [0,1].

    I know that if it is Riemann integrable, sup L(P, f) = inf U(P,f), P being the partition of [0,1]
    or \int f(x)dx (lower integral of f over [0,1]) = \int f(x)dx (upper integral of f over [0,1])

    How do I show if it is or not?
    Notice that on ANY subinterval [a,b]\subset[0,1] that \sup_{x\in[a,b]}f=3 and \inf_{x\in[a,b]}f=1...so what?
    Follow Math Help Forum on Facebook and Google+

  3. #3
    Member thaopanda's Avatar
    Joined
    Sep 2009
    From
    Worcester, Massachusetts
    Posts
    85
    so does that mean it's not Riemann integrable because the sup doesn't equal the inf?
    Follow Math Help Forum on Facebook and Google+

  4. #4
    MHF Contributor Drexel28's Avatar
    Joined
    Nov 2009
    From
    Berkeley, California
    Posts
    4,563
    Thanks
    21
    Quote Originally Posted by thaopanda View Post
    so does that mean it's not Riemann integrable because the sup doesn't equal the inf?
    You tell me. And justify your answer. (I will verify it if you do)
    Follow Math Help Forum on Facebook and Google+

  5. #5
    Member thaopanda's Avatar
    Joined
    Sep 2009
    From
    Worcester, Massachusetts
    Posts
    85
    well.. umm...

    so L(P,f) = \sum m_{k}(x_{k}-x_{k-1}) where m_{k} = inf f(x)

    and U(P,f) = \sum M_{k}(x_{k}-x_{k-1}) where M_{k} = sup f(x)

    since sup f = 3 and inf f = 1,

    L(P,f) = \sum (x_{k}-x_{k-1})
    U(P,f) = \sum 3(x_{k}-x_{k-1})

    and (x_{k}-x_{k-1}) \leq 1 because that's the maximum partition in [0,1]

    so,
    sup L(P,f) \leq 1
    inf U(P,f) \leq 3

    but the sup f = 3... yeah, I'm lost...
    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, 10:19 PM
  2. Is a bounded integrable function square integrable?
    Posted in the Differential Geometry Forum
    Replies: 1
    Last Post: May 28th 2010, 07:26 PM
  3. [SOLVED] f integrable implies f^2 integrable
    Posted in the Differential Geometry Forum
    Replies: 15
    Last Post: June 8th 2009, 11:53 PM
  4. Reimann Sum Help
    Posted in the Calculus Forum
    Replies: 2
    Last Post: February 5th 2009, 12:45 PM
  5. reimann sum
    Posted in the Calculus Forum
    Replies: 1
    Last Post: April 17th 2007, 04:53 PM

Search Tags


/mathhelpforum @mathhelpforum