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Math Help - any example of sequence of continuous function

  1. #1
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    any example of sequence of continuous function

    fn is a sequence of uniformly continuous real-valued functions on R.

    and fn converges pointwise to f. which is continuous but not uniformly continuous.
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  2. #2
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    give example of a continuous function

    f:[0, infty) ----> R is continuous.

    f(x) tends to a finite limit as x-->infty

    must f be uniformly continuous on [0,infty)?

    any counterexample
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  3. #3
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    Quote Originally Posted by szpengchao View Post
    f:[0, infty) ----> R is continuous. f(x) tends to a finite limit as x-->infty must f be uniformly continuous on [0,infty)?
    We can assume that \lim _{x \to \infty } f(x) = L
    In this case, \varepsilon  > 0 \Rightarrow \left( {\exists N} \right)\left[ {x \geqslant N \Rightarrow \left| {f(x) - L} \right| < \frac{\varepsilon }{2}} \right].
    Now I ask you is f uniformly continuous on \left[ {0,N} \right]? (Why or Why Not?)
    So what is your answer?
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  4. #4
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    no

    no, i think. for that function to be uni. continuous on [a,b], we need:

    \forall \epsilon>0, \ \exists \delta>0, \forall x,y\in[0,N], \ |x-y|<\delta\Rightarrow |f(x)-f(y)|<\epsilon

    and f(y) takes value L at  y\rightarrow\infty
    and therefore we cannot have an arbitary small distance between x,y.


    i appreciate your way of "teaching."
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  5. #5
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    If a function is continuous on a compact set, such as [0,N], is it uniformly continuous on that set?
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  6. #6
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    yes

    yes
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