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Pointwise vs Uniform Convergence

  1. May 24, 2007 #1

    Hope this is the right forum, I'm not quite sure myself.
    Anyway, this problem has been bugging me for what seems way too long for such an apparently simple problem.
    It's about the difference between Pointwise and Uniform Convergence (Topology).

    By reading different articles online I've gotten a somewhat good understanding of the basic principles,
    but I do not manage to make it 100% intuitive (which I feel I need to do, in order to "move on").

    One way to formulate my problem is this; according to Wikipedia (and others):
    "In the case of uniform convergence, N can only depend on ε, while in the case of pointwise convergence N may depend on ε and x."
    And my question then would be; but why can't one just look at the domain and define N by some extremum obtained?

    I know my question is somewhat poorly formulated and the fact that I don't manage to formulate it properly is testimony to my incomplete understanding of the subject.
    Still, any help would be greatly appreciated.

  2. jcsd
  3. May 24, 2007 #2

    matt grime

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    Given e (for epsilon) you can try to take the sup over all the N(e,x) over all x. But this could be, and in general will be, infinity, unless it is uniform convergence.
  4. May 24, 2007 #3
    Because this extremum might not go to 0!

    Take the sequence of functions [itex]\left\{f_n\right\}_{n\in\mathbb{N}}[/itex] given by

    [tex]f_n\left(x\right) = \left\{
    1 & \mbox{if } 0 < x < 1/n \\
    0 & \mbox{otherwise}
    \right .[/tex]

    This function converges pointwise to the function [itex]f\left(x\right) = 0[/itex] on [itex]\mathbb{R}[/itex], but not uniformly. Can you see why?
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