Uniform Convergence: Showing It's Not Pointwise Convergent

In summary, the conversation is about a student seeking feedback for studying for finals. They ask how to show that a pointwise convergent sequence of functions is not uniformly convergent and mention the limitations of interchanging limits and derivatives or integrals. They also ask for a counterexample to this concept.
  • #1
happyg1
308
0
Hi,
I'm studying for finals and I just need some feedback.
One of questions MIGHT be:
If I know a sequence of functions is pointwise convergent, how do I show that it's not uniformly convergent?
I think that a pointwise convergent sequence of functions might not converge to a continuous function, although it might.
Also, you can't interchange the limit and the derivitive, or the limit and the integral.
Am I right? Am I missing something?
Any feedback will be appreciated.
Thanks,
CC
 
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  • #2
Find a counterexample. A standard example involves growing steeples, i.e. the nth function is 0 everywhere except on [2n,2n+2], say, and on this interval, it is basically an isoceles triangle with its value at 2n and 2n+2 being 0, and its value at 2n+1 being n, say.
 

FAQ: Uniform Convergence: Showing It's Not Pointwise Convergent

What is uniform convergence?

Uniform convergence is a type of convergence of a sequence of functions, where the rate of convergence is independent of the variable. This means that for any given value, the functions in the sequence become closer and closer to each other at the same rate.

How is uniform convergence different from pointwise convergence?

In pointwise convergence, the functions in the sequence may converge at different rates for different values of the variable. This means that the pointwise limit function may not accurately represent the behavior of the entire sequence. In contrast, uniform convergence ensures that the sequence of functions converges uniformly, providing a more accurate representation of the overall behavior.

How is uniform convergence proven?

To prove uniform convergence, one must show that for any given ε (epsilon), there exists an N (natural number) such that for all n > N, the distance between the function in the sequence and the limit function is less than ε for all values of the variable. This ensures that the rate of convergence is consistent and independent of the variable.

Why is uniform convergence important?

Uniform convergence is important because it guarantees that the limit function accurately represents the entire sequence of functions. This is especially useful in applications where the behavior of the entire sequence is important, such as in numerical analysis and approximation methods.

Can a sequence of functions be both uniformly and pointwise convergent?

Yes, a sequence of functions can be both uniformly and pointwise convergent. In fact, uniform convergence implies pointwise convergence. However, the reverse is not always true, as a sequence can be pointwise convergent but not uniformly convergent.

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