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Math Help - questions on uniform convergence

  1. #1
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    questions on uniform convergence

    if (x_n) is a sequence that converges uniformly to x and f a continuous fonction

    can we say that f(x_n) converges uniformly to f(x)?

    thank you
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  2. #2
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    Quote Originally Posted by deubelte View Post
    if (x_n) is a sequence that converges uniformly to x and f a continuous function.

    can we say that f(x_n) converges uniformly to f(x)?

    thank you
    uniform convergence has meaning for a sequence of functions only! are x_n functions?
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  3. #3
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    ahh yes sorry.
    to be more specific, this is what I have:
    I have a sequence of functions (f_n) that converges uniformly to f.

    Now, let d be the derivative operator. It is continuous.
    I would like to know if
    df_n converge uniformly to df?

    thank you.
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  4. #4
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    Quote Originally Posted by deubelte View Post
    ahh yes sorry.
    to be more specific, this is what I have:
    I have a sequence of functions (f_n) that converges uniformly to f.

    Now, let d be the derivative operator. It is continuous.
    I would like to know if
    df_n converge uniformly to df?

    thank you.
    not necessarily! in order for that to happen a sufficient condition is that the sequence \{f_n' \} converges uniformly too.
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  5. #5
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    Quote Originally Posted by NonCommAlg View Post
    not necessarily! in order for that to happen a sufficient condition is that the sequence \{f_n' \} converges uniformly too.
    well, this is what I want to show.

    (df_n) is the differential operator.

    In fact, f is C^1 in R^n and f_n is a polynomial function that converges uniformly to f. THis is the Weierstrass theorem.

    Can I say that df_n converges uniformly to df??
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