Uniform convergence and continuity

In summary, the theorem states that if all functions f_k are continuous at a fixed point x_0 in the metric space X and f_k converges uniformly to f, then f is also continuous at x_0. However, this is not always the case as demonstrated by the example f_k(x)=x^k on [0,1] where the limit f(x) is not continuous. This example highlights the importance of considering pointwise convergence when determining the continuity of a function.
  • #1
complexnumber
62
0

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?
 
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  • #2
f_k(x)=x^k on [0,1]. What's the limit f(x)?
 
  • #3
Dick said:
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.
 

1. What is uniform convergence?

Uniform convergence is a type of convergence in which the limit function and the sequence of functions approach each other uniformly over a given interval, rather than pointwise. This means that the distance between the limit function and the sequence of functions becomes smaller and smaller at every point in the interval.

2. How is uniform convergence different from pointwise convergence?

In pointwise convergence, the limit function and the sequence of functions approach each other at each individual point. However, in uniform convergence, the convergence occurs over a given interval and is independent of any specific point.

3. What are the criteria for uniform convergence?

To determine if a sequence of functions converges uniformly, we can use the Weierstrass M-test or the Cauchy criterion. The Weierstrass M-test states that if the absolute value of the sequence of functions is less than or equal to a convergent series (M) at every point in the interval, then the sequence of functions converges uniformly. The Cauchy criterion states that if the difference between two terms in the sequence of functions is less than any given positive number (epsilon) at every point in the interval, then the sequence of functions converges uniformly.

4. How does uniform convergence relate to continuity?

If a sequence of continuous functions converges uniformly, then the limit function is also continuous. This means that the continuity of the sequence of functions is preserved in the limit function. In other words, uniform convergence guarantees the continuity of the limit function.

5. Can a sequence of discontinuous functions converge uniformly?

Yes, it is possible for a sequence of discontinuous functions to converge uniformly. This is because uniform convergence is determined by the behavior of the functions over a given interval, rather than at individual points. As long as the distance between the limit function and the sequence of functions becomes smaller and smaller at every point in the interval, the sequence can still converge uniformly.

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