Pointwise vs uniform convergence

In summary, the conversation is discussing the investigation of the pointwise limit of a sequence of functions f_n over [0, 1] and the determination of whether it converges uniformly or not. It is noted that the pointwise limit is f(x) = lim_{n\rightarrow\infty} x^{n} / (1 + x^{n}), and there is some confusion as to whether the sequence converges uniformly due to the discontinuity of f(x). The speaker also questions the validity of \sup\{|f_n(x) - f(x)| \colon x \in [0,1]\} tending to 0 as n tends to infinity.
  • #1
sephiseraph
6
0
Howdy Ho, partner.

I have a series of functions [tex]{f_{n}}[/tex] with [tex]f_{n}(x) := x^{n} / (1 + x^{n})[/tex] and I am investigating the pointwise limit of the sequence [tex]f_{n}[/tex] over [0, 1] to see if it converges uniformly.

I found the pointwise limit f(x) to be [tex]f(x) = lim_{n\rightarrow\infty} x^{n} / (1 + x^{n})[/tex], f(x) = 0 for x in [0, 1), f(x) = 1/2 for x = 1.

My problem here comes in finding out if the sequence converges uniformly or not. Intuition tells me that, since f(x) is not continuous, there cannot be uniform convergence on [0, 1]. However we also have

[tex]sup\left\{|f_{n}(x) - f(x)| : x \in \left[0, 1\right]\right\} = sup\left\{x^{n} / (1 + x^{n}) : x \in \left[0, 1\right)\right\}[/tex] which tends to 0 as n tends to infinity, which is a sufficient condition for uniform convergence. Where have I gone wrong here or how do I make sense of this?
 
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  • #2
Are you sure [itex]\sup\{|f_n(x) - f(x)| \colon x \in [0,1]\}[/itex] tends to 0?
 

1. What is the difference between pointwise and uniform convergence?

Pointwise convergence refers to a sequence of functions converging to a limit function at each individual point in the domain. Uniform convergence, on the other hand, means that the sequence of functions converges to the limit function uniformly, meaning that the difference between the functions becomes arbitrarily small for all points in the domain.

2. Which type of convergence is stronger, pointwise or uniform?

Uniform convergence is considered to be stronger than pointwise convergence because it guarantees convergence at every point in the domain, whereas pointwise convergence only guarantees convergence at each individual point.

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

Yes, it is possible for a sequence of functions to be both pointwise and uniformly convergent. This occurs when the sequence of functions converges to the same limit function for both types of convergence.

4. How is the rate of convergence different between pointwise and uniform convergence?

The rate of convergence is typically faster for uniform convergence compared to pointwise convergence. This is because uniform convergence requires the sequence of functions to converge to the limit function at every point simultaneously, whereas pointwise convergence only requires convergence at each point individually.

5. In what situations would you use pointwise convergence over uniform convergence?

Pointwise convergence is useful when studying individual behavior or properties of a function at specific points in the domain. It can also be used to prove pointwise properties of a function, such as continuity or differentiability. However, in most practical applications, uniform convergence is preferred as it guarantees convergence over a larger portion of the domain.

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