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Homework Help: Uniform convergence and continuity

  1. Apr 2, 2010 #1
    1. The problem statement, all variables and given/known data

    Theorem:

    Let [tex](X,d_X),(Y,d_Y)[/tex] be metric spaces and let [tex]f_k : X \to Y[/tex], [tex]f :
    X \to Y[/tex] be functions such that
    1. [tex]f_k[/tex] is continuous at fixed [tex]x_0 \in X[/tex] for all [tex]k \in \mathbb{N}[/tex]
    2. [tex]f_k \to f[/tex] uniformly
    then [tex]f[/tex] is continuous at [tex]x_0[/tex].

    2. Relevant equations

    If all [tex]f_k[/tex] are continuous on [tex]X[/tex] and [tex]f_k \to f[/tex] pointwise, then [tex]f[/tex] need not be continuous. Why?

    3. The attempt at a solution

    I really can't think of an example. Can someone please explain to me why this is so or give me an example?
     
  2. jcsd
  3. Apr 2, 2010 #2

    Dick

    User Avatar
    Science Advisor
    Homework Helper

    f_k(x)=x^k on [0,1]. What's the limit f(x)?
     
  4. Apr 2, 2010 #3
    The limit [tex]f(x)[/tex] is
    [tex]
    \begin{align*}
    f(x) =& 0 \text{ if } 0 \leq x < 1 \\
    f(x) =& 1 \text{ if } x = 1
    \end{align*}
    [/tex]
    which is not continuous.

    But every [tex]f_k(x) = x^k[/tex] is continuous and converges pointwise to [tex]f[/tex].

    Thank you so much. I never thought about functions like this.
     
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