1. Limited time only! Sign up for a free 30min personal tutor trial with Chegg Tutors
    Dismiss Notice
Dismiss Notice
Join Physics Forums Today!
The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

Homework Help: Uniform convergence and continuity

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


    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


    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
    f(x) =& 0 \text{ if } 0 \leq x < 1 \\
    f(x) =& 1 \text{ if } x = 1
    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.
Share this great discussion with others via Reddit, Google+, Twitter, or Facebook