Results 1 to 2 of 2

Thread: Uniform Continuous and Uniform Convergence

  1. #1
    Member
    Joined
    Jan 2007
    Posts
    114

    Uniform Continuous and Uniform Convergence

    24.13 Prove that if (fn) is a sequence of uniformly continuous functions on an interval (a,b), and if fn -> f uniformly on (a,b), the f is also uniformly continuous on (a,b). Hint use eps/3 argument.

    I don't see what I need to change from the "limit of continuous functions is continuous" theorem. At the end of that proof it concludes f is continous at x0. I guess I'm not sure how to expand that to uniform continuity.

    Thanks
    Follow Math Help Forum on Facebook and Google+

  2. #2
    MHF Contributor

    Joined
    Aug 2006
    Posts
    21,736
    Thanks
    2811
    Awards
    1
    If $\displaystyle \varepsilon > 0$ then $\displaystyle \left( {\exists N} \right)\left( {\forall x \in (a,b)} \right)\left[ {n \ge N \Rightarrow \quad \left| {f_N (x) - f(x)} \right| < \frac{\varepsilon }{3}} \right]$ from uniform convergence.

    Now from uniform continuity $\displaystyle \left( {\exists \delta > 0} \right)\left[ {\left| {x - y} \right| < \delta \Rightarrow \quad \left| {f_N (x) - f_N (y)} \right| < \frac{\varepsilon }{3}} \right]$.

    $\displaystyle \begin{array}{l}
    \\
    \left| {f(x) - f(y)} \right| \le \left| {f(x) - f_N (x)} \right| + \left| {f_N (x) - f_N (y)} \right| + \left| {f_N (y) - f(y)} \right| \\
    \end{array}$
    Follow Math Help Forum on Facebook and Google+

Similar Math Help Forum Discussions

  1. Replies: 1
    Last Post: Oct 31st 2010, 07:09 PM
  2. Proving a function is continuous using uniform convergence
    Posted in the Differential Geometry Forum
    Replies: 1
    Last Post: Apr 12th 2010, 09:00 PM
  3. Continuous uniform distribution
    Posted in the Advanced Statistics Forum
    Replies: 6
    Last Post: Feb 24th 2009, 04:20 AM
  4. Not Uniform Continuous?
    Posted in the Calculus Forum
    Replies: 5
    Last Post: Oct 16th 2008, 11:06 AM
  5. Uniform Continuous?
    Posted in the Calculus Forum
    Replies: 1
    Last Post: Oct 12th 2008, 02:57 AM

Search tags for this page

Search Tags


/mathhelpforum @mathhelpforum