Does pointwise convergence guarantee the intermediate value property?

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In summary, the conversation discusses the convergence of a sequence of functions and whether the intermediate value property holds for the limiting function. It is stated that pointwise convergence is not enough for the intermediate value property to hold, and that uniform convergence is necessary. There is also a discussion about continuous functions and their convergence, with examples given to illustrate the importance of uniform convergence. Finally, it is mentioned that the intermediate value property may not hold without uniform convergence, and an example is provided to support this claim.
  • #1
Treadstone 71
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If a sequence of functions f_n converges pointwise to a bounded function f, does f have the intermediate value property? If not, are there some conditions that will make it so?
 
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  • #2
No. Consider f_n = f for all f and f(x) = 0 for x<1 and 1 for x geq 1.

The intermediate value property holds for continuous functions. But even then pointwise convergence is not enough. Consider f_n=x^n on [0,1]. The limiting function is f(x)=0 for x in [0,1) and f(1)=1.

However, the limit of a sequence of continuous functions will be continuous if you have uniform convergence.
 
  • #3
Treadstone 71 said:
If a sequence of functions f_n converges pointwise to a bounded function f, does f have the intermediate value property? If not, are there some conditions that will make it so?

As stated this is trivially false. Surely you mean f_n to be continuous at the very least.
 
  • #4
I don't have uniform convergence. But suppose (f_n) is an increasing sequence of continuous functions on R. Suppose (f_n) converges pointwise to f, then {x in R : f(x)>a} must be open, and I can show this if I had the intermediate value property.
 
  • #5
Actually I don't think this is true without uniform convergence. I can think of functions that satisfy the intermediate value property and yet {x in R : f(x)>a} is not open, even closed. I'll repost in the homework section.
 
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  • #6
Let fn(x)= 0 is x<0, nx if 0<= x<= 1/n, 1 if x> 1/n.

Each such f is continuous but the sequence converges pointwise to
f(x)= 0 if x<= 0, 1 if x> 0 which is not continuous and does not satisfy the intermediate value property.
 

What is pointwise convergence?

Pointwise convergence is a concept in mathematics and science that describes the behavior of a sequence of functions. It means that as the index of the sequence increases, the value of the function at a particular point will approach a specific limit.

How is pointwise convergence different from uniform convergence?

Pointwise convergence and uniform convergence are two types of convergence that describe the behavior of sequences of functions. The main difference between them is that pointwise convergence focuses on individual points, while uniform convergence considers the behavior of the entire function. In pointwise convergence, the limit of the function at a specific point may vary as the index increases, while in uniform convergence, the limit is the same for all points in the domain.

What is the significance of pointwise convergence in scientific research?

Pointwise convergence is an essential concept in scientific research, particularly in fields such as physics, engineering, and economics. It allows scientists to analyze the behavior of sequences of functions and make predictions about their limits. Pointwise convergence is also a crucial tool in the development of mathematical models and understanding real-world phenomena.

How is pointwise convergence tested or verified?

In mathematics and science, pointwise convergence is typically verified by using mathematical proofs and techniques such as the epsilon-delta method. This method involves choosing a value of epsilon (ε) and showing that for any delta (δ) greater than zero, the value of the function at a particular point will be within ε of the limit when the index of the sequence is greater than δ.

Can a sequence of functions converge pointwise but not uniformly?

Yes, a sequence of functions can converge pointwise but not uniformly. This means that the limit of the function at a specific point exists, but the sequence of functions does not converge to the same limit for all points in the domain. This can happen when the rate of convergence varies at different points, leading to different limits for each point.

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