Results 1 to 2 of 2

Math Help - prove...

  1. #1
    Member
    Joined
    Aug 2008
    Posts
    172

    prove...

    prove that if f is continuous on [a,b] with  f(x) \geq 0 \ , \forall x \in [a,b] prove that if g is strictly increasing on [a,b]
    with  \int_a^b f . dg =0 then f(x)=0  \forall x \in [a,b]
    Follow Math Help Forum on Facebook and Google+

  2. #2
    MHF Contributor

    Joined
    Aug 2008
    From
    Paris, France
    Posts
    1,174
    Quote Originally Posted by flower3 View Post
    prove that if f is continuous on [a,b] with  f(x) \geq 0 \ , \forall x \in [a,b] prove that if g is strictly increasing on [a,b]
    with  \int_a^b f . dg =0 then f(x)=0  \forall x \in [a,b]
    The proof is the same as for usual Riemann integral. Procede by contradiction: assume that f is not identically zero on [a,b]. Using continuity, prove that there exists \epsilon>0 and a\leq u<v\leq b such that f(x)\geq \epsilon when x\in[u,v]. Then, justify the following: \int_a^b f\, dg\geq \int_u^v f\, dg\geq \epsilon \int_u^v dg=\epsilon(g(v)-g(u))>0. And conclude.
    Follow Math Help Forum on Facebook and Google+

Similar Math Help Forum Discussions

  1. Prove a(AB)=(aA)B=A(aB) ..
    Posted in the Advanced Algebra Forum
    Replies: 1
    Last Post: September 29th 2010, 05:14 AM
  2. Prove: f is one-to-one iff f is onto
    Posted in the Discrete Math Forum
    Replies: 12
    Last Post: June 25th 2010, 11:02 AM
  3. Replies: 2
    Last Post: August 28th 2009, 03:59 AM
  4. Please prove
    Posted in the Differential Geometry Forum
    Replies: 2
    Last Post: April 7th 2009, 02:58 PM
  5. Prove this .
    Posted in the Math Topics Forum
    Replies: 5
    Last Post: February 18th 2009, 05:09 AM

Search Tags


/mathhelpforum @mathhelpforum