• Support PF! Buy your school textbooks, materials and every day products Here!

Accumulation points of the domain of a function

  • #1
let a function f: A -> R^m where A is a subset of R^n. if a is an accumulation point of the domain of f, the set A, then f(A) is an accumulation point of the set R^m.

i'm not sure if this statement is true or not but if it is, how would I prove this result? i was thinking of using the fact that if a function is continuous and the sequence x_n approaches x, then f(x_n) approaches f(x). but the function need not be continuous for a limit to exist.

the question i'm trying to answer for myself is that, if a is not an accumulation point of the domain of a function then the limit as x approaches a is not well defined since it could be any number you choose and you could prove the result vacuously. so if a is an accumulation point of the domain does that necessarily mean that f(a) is an accumulation point as well?
 

Answers and Replies

  • #2
jgens
Gold Member
1,580
49
How about we consider [itex]f:\bbold{R} \to \bbold{R}[/itex] defined by [itex]f(x) = 0[/itex]? Take an accumulation point in the domain of [itex]f[/itex] and see if you find an accumulation point in the image set.
 
  • #3
ah i see. in this case since 0 is the only possible point in the codomain of f(x), f(A) is not always an accumulation point when A is an accumulation point.

are there cases where having f(A) be an accumulation point would be useful? as in proving some theorem or leading to some other concepts? or does having an accumulation point at f(A), given A is an accumulation point, not have much further significance whatsoever?
 

Related Threads on Accumulation points of the domain of a function

Replies
0
Views
1K
  • Last Post
Replies
2
Views
7K
  • Last Post
Replies
2
Views
8K
Replies
1
Views
2K
Replies
8
Views
1K
Replies
5
Views
5K
  • Last Post
Replies
1
Views
3K
  • Last Post
Replies
6
Views
3K
  • Last Post
Replies
2
Views
869
Top