Dismiss Notice
Join Physics Forums Today!
The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

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.
     
Share this great discussion with others via Reddit, Google+, Twitter, or Facebook