Results 1 to 3 of 3

Math Help - Prove that if f is integrable then kf is integrable.

  1. #1
    Junior Member
    Joined
    Dec 2009
    Posts
    30

    Wink Prove that if f is integrable then kf is integrable.

    Okay so this makes sense because if the integral of f exists then kf should exist if k is an element in the reals.

    okay so there is a theorem in my book that says:
    Let a,b, and k be real numbers. a< b. Let f and g be a real-valued functions that are Riemann integrable on [a,b]. Then integral b to a of kf equals k* integral b to a of f.

    So i start out by saying e>0 and that there is a delta>0 such that ||P||<delta.

    ||P|| = max(xi - x(i-1))<delta

    I just can't figure out how to incorporate a real number in when I have been solving proofs like Prove (f+g) = (f)+(g) when f and g are both functions and greater than or equal to zero. When i can solve by using inf and sup to prove ...... can i do that here as well?

    Any help would be appreciated .......

    I will continue looking at the problem
    Follow Math Help Forum on Facebook and Google+

  2. #2
    Member
    Joined
    Aug 2009
    Posts
    130
    If f is integrable then  \int_a^b f(x) dx = \lim_{h \rightarrow 0} h[f(a+h) + f(a+2h) + \cdots + f(a+nh)] where  h = \frac{b-a}{n} and  a + nh = b .

    You also know that  \lim_{h \rightarrow 0} k h[f(a+h) + f(a+2h) + \cdots + f(a+nh)] = k \lim_{h \rightarrow 0} h[f(a+h) + f(a+2h) + \cdots + f(a+nh)] right?
    Follow Math Help Forum on Facebook and Google+

  3. #3
    MHF Contributor Drexel28's Avatar
    Joined
    Nov 2009
    From
    Berkeley, California
    Posts
    4,563
    Thanks
    21
    Quote Originally Posted by derek walcott View Post
    Okay so this makes sense because if the integral of f exists then kf should exist if k is an element in the reals.

    okay so there is a theorem in my book that says:
    Let a,b, and k be real numbers. a< b. Let f and g be a real-valued functions that are Riemann integrable on [a,b]. Then integral b to a of kf equals k* integral b to a of f.

    So i start out by saying e>0 and that there is a delta>0 such that ||P||<delta.

    ||P|| = max(xi - x(i-1))<delta

    I just can't figure out how to incorporate a real number in when I have been solving proofs like Prove (f+g) = (f)+(g) when f and g are both functions and greater than or equal to zero. When i can solve by using inf and sup to prove ...... can i do that here as well?

    Any help would be appreciated .......

    I will continue looking at the problem
    I will note four things and you put them together.

    1. \sup_{x\in[x_{j-1},x_j]}\text{ }kf(x)=k\sup_{x\in[x_{j-1},x_j]}\text{ }f(x) (the same for infimums). Prove this.

    2. From this we see that U(P,kf)=\sum_{j=1}^{n}\left(\sup_{x\in[x_{j-1},x_j]}\text{ }kf(x)\right)\Delta x_j= \sum_{j=1}^{n}k\sup_{x\in[x_{j-1},x_j]}\text{ }f(x)\cdot \Delta x_j=k\sum_{j=1}^{n}\sup_{x\in[x_{j-1},x_j]}\text{ }f(x)\cdot \Delta x_j=k U(P,f)

    Similarly L(P,kf)=kL(P,f)

    3. Using this we see that U(P,kf)-L(P,kf)=k\left(U(P,f)-L(P,f)\right). (think the \varepsilon definition of integrability).


    This proves integrability.

    Now, we know that \int_a^b\text{ }kf=\sup_{P\in\mathcal{P}}\text{ }L(P,kf)=\sup_{P\in\mathcal{P}}\left(k\cdot L(P,f)\right).

    Now apply the ideas of 1. to finish.
    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. Prove x^2 is integrable
    Posted in the Differential Geometry Forum
    Replies: 2
    Last Post: November 6th 2009, 08:22 PM
  4. [SOLVED] f integrable implies f^2 integrable
    Posted in the Differential Geometry Forum
    Replies: 15
    Last Post: June 8th 2009, 11:53 PM
  5. Prove exp(-x) / x is integrable on (0,1)
    Posted in the Calculus Forum
    Replies: 10
    Last Post: February 21st 2009, 03:18 PM

Search Tags


/mathhelpforum @mathhelpforum