Proving Uni.Conv. of Fn to F on (0,1): Questions & Answers

In summary, the question is about proving that if a sequence is uniformly continuous on (0,1) and uniformly converges to a function on (0,1), then the function is also uniformly continuous on (0,1). The proposed proof involves using the continuous extension theorem and proving uniform convergence on [0,1]. The asker is looking for a counterexample or a proof using a standard $\varepsilon/3$ argument.
  • #1
bw0young0math
27
0
Hello, today I have a question.

If uni.cont. function sequence fn on (0,1) is uni.conv. to f on (0,1), then f is uni.cont. on (0,1).

The above is true.
The wonder I have is...May I prove the above by this way?

This way: fn is uniform continuous on (0,1). So Fn is continuous on [0,1] (by continuous extention theorem.)
If I can prove Fn is uni. convergent to F on [0,1], (since Fn is cont. on [0,1]) F is cont. on [0,1] and f is uni. cont. on (0,1).

is this way true?
IF it is true, how can I prove Fn is uni.conv. to F on [0,1]?
IF not, could you show me some counter example?Please...(Sadface)(Sadface)(Sadface)
 
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  • #2
I think you can use a standard $\varepsilon/3$ argument to prove this. Write out what uniform continuity and uniform convergence mean, using $\epsilon/3$ as the RHS. Then use the Triangle Inequality to finish.
 

FAQ: Proving Uni.Conv. of Fn to F on (0,1): Questions & Answers

What is the definition of uniform convergence?

Uniform convergence is a type of convergence of a sequence of functions, where the limit function is approached at a uniform rate by each individual function in the sequence. It means that for any given epsilon, there exists a natural number N such that for all n greater than or equal to N, the distance between the limit function and the n-th function is less than epsilon at all points in the domain.

How do you prove uniform convergence of a sequence of functions?

To prove uniform convergence of a sequence of functions, you must show that for any given epsilon, there exists a natural number N such that for all n greater than or equal to N, the distance between the limit function and the n-th function is less than epsilon at all points in the domain. This can be done by using the definition of uniform convergence and using techniques such as the Cauchy criterion or the Weierstrass M-test.

What is the importance of proving uniform convergence?

Proving uniform convergence is important because it guarantees that the limit function is approached by each individual function in the sequence at a uniform rate. This is important in many areas of mathematics and science, such as in the study of series, differential equations, and numerical analysis.

Can a sequence of functions converge pointwise but not uniformly?

Yes, it is possible for a sequence of functions to converge pointwise but not uniformly. Pointwise convergence only requires that for each point in the domain, the function values approach the limit function as n approaches infinity. However, for uniform convergence, the rate of convergence must be the same at all points in the domain.

What is the significance of proving uniform convergence on the interval (0,1)?

Proving uniform convergence on the interval (0,1) is significant because it is a commonly used interval in mathematics and science. It also has important applications in the study of calculus and real analysis, as it is the interval in which many fundamental theorems and concepts are defined and studied.

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