Results 1 to 6 of 6

Thread: First fundamental theorem, question about continuity

  1. #1
    Member Mollier's Avatar
    Joined
    Nov 2009
    From
    Norway
    Posts
    234
    Awards
    1

    First fundamental theorem, question about continuity

    Hi,

    I was reading the proof for the first fundamental theorem in Apostol, and one part of it says the following.

    Continuity of $\displaystyle f$ at $\displaystyle x$ tells us that, if $\displaystyle \epsilon$ is given, there is a positive $\displaystyle \delta$ such that

    $\displaystyle |f(t)-f(x)|<\frac{1}{2}\epsilon$

    whenever

    $\displaystyle x-\delta < t < x+\delta$

    I do not see how $\displaystyle |f(t)-f(x)|$ must be less than $\displaystyle \frac{1}{2}\epsilon$. I do see why it has to be less than $\displaystyle \epsilon$ though..

    Any hints?

    Thanks.
    Follow Math Help Forum on Facebook and Google+

  2. #2
    Super Member
    Joined
    Aug 2009
    From
    Israel
    Posts
    976
    It's simply a matter of formalism - the author chose to take $\displaystyle \epsilon_0 = \frac{1}{2} \epsilon$ and used $\displaystyle \epsilon_0$ in the definition of continuity.
    Follow Math Help Forum on Facebook and Google+

  3. #3
    Member Mollier's Avatar
    Joined
    Nov 2009
    From
    Norway
    Posts
    234
    Awards
    1
    So he just chooses a smaller neighborhood?
    Follow Math Help Forum on Facebook and Google+

  4. #4
    Super Member
    Joined
    Aug 2009
    From
    Israel
    Posts
    976
    Well, it doesn't really matter. The proof would've worked as well, had he taken $\displaystyle \epsilon$ instead of $\displaystyle \frac{1}{2} \epsilon$. The only difference would be that in the end, he would have $\displaystyle | \text{something} | < 2 \epsilon$, and by taking $\displaystyle \epsilon_0 = \frac{1}{2} \epsilon$, he will end up with $\displaystyle | \text{something} |< \epsilon$.
    Follow Math Help Forum on Facebook and Google+

  5. #5
    Member Mollier's Avatar
    Joined
    Nov 2009
    From
    Norway
    Posts
    234
    Awards
    1
    Ah, he's really thinking ahead then. Must be a good chess player
    Thanks mate.
    Follow Math Help Forum on Facebook and Google+

  6. #6
    MHF Contributor

    Joined
    Apr 2005
    Posts
    19,781
    Thanks
    3030
    Quote Originally Posted by Mollier View Post
    Ah, he's really thinking ahead then. Must be a good chess player
    Thanks mate.
    Actually, most proofs like this are developed by taking "$\displaystyle \epsilon$", seeing at the end that it would have been better to use "$\displaystyle \frac{1}{2}\epsilon$", then going back and changing it- which you can't do in chess.
    Follow Math Help Forum on Facebook and Google+

Similar Math Help Forum Discussions

  1. Replies: 0
    Last Post: Dec 11th 2011, 08:00 PM
  2. Replies: 8
    Last Post: Nov 29th 2010, 12:40 PM
  3. easy question involving fundamental theorem
    Posted in the Calculus Forum
    Replies: 9
    Last Post: Dec 17th 2009, 12:47 AM
  4. Fundamental theorem
    Posted in the Calculus Forum
    Replies: 1
    Last Post: Nov 28th 2007, 12:40 PM
  5. Replies: 2
    Last Post: Jun 14th 2007, 06:35 AM

Search Tags


/mathhelpforum @mathhelpforum