A question about limit superior for function

  • Thread starter zzzhhh
  • Start date
  • #1
40
1
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!
 

Answers and Replies

  • #2
mathman
Science Advisor
7,942
496
It is not true as you stated it. The limsup (x->a) ≤ c for any sequence and = c for at least one sequence.
 
  • #3
40
1
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
mathman
Science Advisor
7,942
496
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
40
1
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
Office_Shredder
Staff Emeritus
Science Advisor
Gold Member
4,550
584
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
 

Related Threads on A question about limit superior for function

  • Last Post
Replies
4
Views
3K
  • Last Post
Replies
1
Views
6K
  • Last Post
Replies
3
Views
7K
Replies
2
Views
3K
Replies
5
Views
3K
Replies
3
Views
4K
Replies
8
Views
10K
  • Last Post
Replies
3
Views
2K
  • Last Post
Replies
7
Views
1K
  • Last Post
Replies
9
Views
2K
Top