A question about limit superior for function

In summary, the problem of whether the limit of a function can be converted to the limit of a sequence still holds for the limit superior of a function. However, it is not always true and only holds for continuous functions at the point.
  • #1
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It is well known that limit of function can be converted to limit of sequence. I wonder if it still holds for limit superior of function. This problem is formulated as follows: For function [tex]f:\mathbb R\rightarrow\mathbb R[/tex] and [tex]a\in\mathbb R[/tex], define [tex]{\lim\sup}\limits_{x\to a}f(x)[/tex] to be [tex]\inf\limits_{\delta>0}(\sup\limits_{0<|x-a|<\delta}f(x))[/tex]. Can we have [tex]{\lim\sup}\limits_{x\to a}f(x)=c[/tex] iff [tex]{\lim\sup}\limits_{n\to\infty}f(x_n)=c[/tex] for any sequence [tex]<x_n>[/tex] satisfying 1)[tex]x_n\in\mathbb R[/tex], 2)[tex]x_n\to a[/tex] and 3)[tex]x_n\ne a[/tex]. I have no idea how to prove it, can you help me? Thanks!
 
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  • #2
It is not true as you stated it. The limsup (x->a) ≤ c for any sequence and = c for at least one sequence.
 
  • #3
mathman said:
It is not true as you stated it. The limsup (x->a) ≤ c for any sequence and = c for at least one sequence.
I cannot understand your reply, could you please explain in more detail? Thanks.
 
  • #4
Example f(x)=1 for x rational, f(x)=0 for x irrational. Let a=0, limsup(x->a) f(x)=1. Take any sequence (xk) of irrational numbers converging to a, limsup f(xk)=0.
 
  • #5
A great example, I got it! Thank you!
My original intention is to try to establish the statement "both lim sup f(x) and lim inf f(x) exist and equal c (possibly [tex]\pm\infty[/tex]) iff lim f(x) exists and equals c" from analogical statement for sequence. But now this approach is not feasible. I then proved the above statement for functions by definition. Thank you again, mathman!
 
  • #6
It is well known that limit of function can be converted to limit of sequenc

This is only true for functions that are continuous at the point. So it's not surprising that a limsup styled in the same manner would fail for a function that is everywhere discontinuous
 

1. What is the definition of limit superior for a function?

The limit superior for a function, denoted as lim sup f(x), is the highest possible limit that the function can reach as the input approaches a certain value. It represents the upper bound of the function's limit as the input approaches the specified value.

2. How is limit superior different from limit inferior?

Limit superior is the highest possible limit that a function can reach, while limit inferior is the lowest possible limit. In other words, limit superior represents the upper bound of the function's limit, and limit inferior represents the lower bound.

3. What does it mean if the limit superior of a function does not exist?

If the limit superior of a function does not exist, it means that the function has no upper bound as the input approaches the specified value. This could be due to the function oscillating or approaching infinity as the input gets closer to the specified value.

4. Can the limit superior of a function be equal to its limit at a certain point?

Yes, it is possible for the limit superior of a function to be equal to its limit at a certain point. This occurs when the function has a finite limit at that point and no other values of the function approach that limit from above.

5. How can limit superior be used in real-world applications?

Limit superior can be used in many real-world applications, such as analyzing the growth rate of a population or the speed of an object. It can also be used in mathematical proofs to establish the existence of a limit for a given function.

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