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Uniform convergence and continuity

  • #1

Homework Statement



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].

Homework 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?

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?
 

Answers and Replies

  • #2
Dick
Science Advisor
Homework Helper
26,258
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f_k(x)=x^k on [0,1]. What's the limit f(x)?
 
  • #3
f_k(x)=x^k on [0,1]. What's the limit f(x)?
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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