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Limits of composite functions

  1. Oct 4, 2007 #1
    When is the following true?

    [tex]f(lim_{n\rightarrow\infty}\ g_{n}(x))[/tex] =

    [tex]lim_{n\rightarrow \infty}\ f(g_{n}(x))[/tex]

    Does anyone know of a textbook that discusses this?
     
  2. jcsd
  3. Oct 5, 2007 #2
    Assuming f and g_n are complex-valued, iff f is continuous.
     
  4. Oct 5, 2007 #3
    But what if f is the following function:

    [tex]f(g(x)) = \int^{b}_{a} g(x) dx[/tex]

    If that's the case,

    [tex]f(lim_{n\rightarrow\infty} g_{n}(x)) = \int^{b}_{a} lim_{n\rightarrow\infty}g_{n}(x) dx[/tex]

    and

    [tex]lim_{n\rightarrow\infty} f(g_{n}(x)) = lim_{n\rightarrow\infty} \int^{b}_{a} g_{n}(x) dx[/tex]

    These two are equal only when [tex]g_{n}(x)[/tex] is uniformly convergent.
     
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