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Math Help - uniform convergence

  1. #1
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    uniform convergence

    problem: if a sequence of continuous functions fn from X to Y converges uniformly to f on every compact subset of X, i need to show that f is continuous.


    it might be useful to use the fact that given a point x0, if for every sequence xk converging to x0 we have f(xk) converges to x0, then f is continuous.

    also, if a metric space is X is compact itself, then f is continuous under the conditions in the problem (but of course X can't be assumed to be compact in the problem)

    anyway, i just need a place to start because i really don't know what to make of the whole "every compact subset" thing--does it mean that there is potentially a separate delta (or N) for each compact subset or that the same delta may be used for any compact subset of X? i am confused...
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  2. #2
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    Quote Originally Posted by jimmybuffet View Post
    problem: if a sequence of continuous functions fn from X to Y converges uniformly to f on every compact subset of X, i need to show that f is continuous.
    If f_n: X\mapsto Y are all continous.
    Consider,
    |f(x) - f(x_0)| \leq |f(x) - f_n(x)|+|f_n(x)-f_n(x_0)|+|f_n(x_0)-f(x_0)|
    All three can be made small.
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  3. #3
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    thank you, that is helpful. btw either you're really good at pacman, or you're the only one who plays it. jk nice awards there.
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  4. #4
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    yeah...

    well i still haven't figured out how to use the idea of compactness here, so maybe i'll just have to face up to all the evidence and admit that i am an idiot--anyway, can anyone help?
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  5. #5
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    Quote Originally Posted by jimmybuffet View Post
    well i still haven't figured out how to use the idea of compactness here, so maybe i'll just have to face up to all the evidence and admit that i am an idiot--anyway, can anyone help?
    Just note that \left[ {x_0  - 1,x_0  + 1} \right] is a compact subset of \Re.
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  6. #6
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    okay

    it is actually supposed to be f: X to Y where X and Y are arbitrary metric spaces, but since R isn't compact, i'll try to generalize from what you say (closed neighborhood or something? but not every closed and bounded set is compact i am told)
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