Results 1 to 10 of 10

Math Help - Riemann integration and Lipschitz

  1. #1
    Member
    Joined
    Sep 2009
    Posts
    104

    Riemann integration and Lipschitz

    I need to show that, given f:[a,b]-->R where f is Riemann integrable, and F:[a,b]-->R defined by F(x)= \int^x_a f(t)dt prove that F is Lipschitz... kind of stuck on how to approach this.
    Any help would be much appreciated, thanks
    Follow Math Help Forum on Facebook and Google+

  2. #2
    Senior Member roninpro's Avatar
    Joined
    Nov 2009
    Posts
    485
    Hello.

    Recall that F is Lipschitz continuous if there exists a fixed K>0 such that for every x,y\in \mathbb{R}, |F(x)-F(y)|\leq K |x-y|.

    Can you think of a way to find such a K?
    Follow Math Help Forum on Facebook and Google+

  3. #3
    Member
    Joined
    Sep 2009
    Posts
    104
    well, I know that for some c in [a,b], F'(c)=f(c), not sure where this gets me though...
    Follow Math Help Forum on Facebook and Google+

  4. #4
    Math Engineering Student
    Krizalid's Avatar
    Joined
    Mar 2007
    From
    Santiago, Chile
    Posts
    3,654
    Thanks
    14
    F is differentiable as long as f were given as continuous, but f was given as an integrable function, so it's bounded.
    Follow Math Help Forum on Facebook and Google+

  5. #5
    Senior Member roninpro's Avatar
    Joined
    Nov 2009
    Posts
    485
    I recommend using the Mean Value Theorem.

    For every x,y\in [a,b] there exists c\in (a,b) such that \frac{F(x)-F(y)}{x-y}=F'(c). What happens when you maximize F'?
    Follow Math Help Forum on Facebook and Google+

  6. #6
    Member
    Joined
    Sep 2009
    Posts
    104
    not sure what you mean by maximizing F'. Do you mean set F'(c)=0?
    Follow Math Help Forum on Facebook and Google+

  7. #7
    Super Member
    Joined
    Apr 2009
    From
    México
    Posts
    721
    Quote Originally Posted by roninpro View Post
    I recommend using the Mean Value Theorem.

    For every x,y\in [a,b] there exists c\in (a,b) such that \frac{F(x)-F(y)}{x-y}=F'(c). What happens when you maximize F'?
    As Krizalid said, this works if f is continous, but it's only assumed integrable. Use the fact that it's bounded and that \vert \int_{a}^{b} f(t)dt \vert \leq \int_{a}^{b} \vert f(t) \vert dt
    Follow Math Help Forum on Facebook and Google+

  8. #8
    Member
    Joined
    Sep 2009
    Posts
    104
    I still dont see it,
    so from your point, I gathered that |F(x)|<= integral(f(t)) but I dont see how this will lead me to the lipschitz condition
    Follow Math Help Forum on Facebook and Google+

  9. #9
    Super Member
    Joined
    Apr 2009
    From
    México
    Posts
    721
    \vert F(x)-F(y)\vert =\vert \int_{x}^{y} f(t)dt \vert \leq M\vert x-y \vert where \vert f(t)\vert \leq M.
    Follow Math Help Forum on Facebook and Google+

  10. #10
    Member
    Joined
    Sep 2009
    Posts
    104
    Thank you Jose, sorry to make you spell it all out like that, I guess I was ignoring the boundedness.
    Follow Math Help Forum on Facebook and Google+

Similar Math Help Forum Discussions

  1. Riemann Integration
    Posted in the Calculus Forum
    Replies: 7
    Last Post: May 16th 2011, 12:44 PM
  2. Riemann Sum/Integration
    Posted in the Calculus Forum
    Replies: 3
    Last Post: February 17th 2010, 06:17 PM
  3. riemann integration
    Posted in the Differential Geometry Forum
    Replies: 0
    Last Post: December 6th 2009, 04:29 PM
  4. riemann integration help
    Posted in the Differential Geometry Forum
    Replies: 5
    Last Post: December 6th 2009, 03:59 PM
  5. Riemann Integration
    Posted in the Calculus Forum
    Replies: 5
    Last Post: November 18th 2009, 12:29 AM

Search Tags


/mathhelpforum @mathhelpforum