Confused about continuity and limits

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Discussion Overview

The discussion revolves around the concept of limits in mathematical analysis, specifically focusing on the definition of limits involving neighborhoods and accumulation points. Participants explore the nuances of continuity, the role of deleted neighborhoods, and the implications of these definitions in various contexts.

Discussion Character

  • Exploratory
  • Technical explanation
  • Conceptual clarification
  • Debate/contested

Main Points Raised

  • One participant expresses confusion about the theorem involving limits and neighborhoods, questioning why a deleted neighborhood is used instead of a standard neighborhood.
  • Another participant suggests that the use of a deleted neighborhood is necessary because the point c may not be included in the domain D, which could affect the limit's definition.
  • A later reply points out that even if c is not in D, the limit can still be defined correctly, emphasizing that the function may not be continuous at c.
  • One participant provides an example function to illustrate the concept, noting that the limit can still be defined even when the function value at the point differs from the limit.
  • Another participant advises looking for the unusual aspects of the proof to better understand the limit's definition and its implications.
  • One participant acknowledges their misunderstanding and clarifies that they were initially focused on continuity rather than the limit itself.

Areas of Agreement / Disagreement

Participants do not reach a consensus on the initial confusion regarding the use of deleted neighborhoods, but they engage in a constructive dialogue that clarifies various aspects of the limit definition. Some participants agree on the importance of focusing on limits rather than continuity at the point.

Contextual Notes

Some assumptions about the type of space being discussed are not fully clarified, which may affect the understanding of the limit and neighborhood definitions.

Who May Find This Useful

Readers interested in mathematical analysis, particularly those grappling with the concepts of limits, continuity, and neighborhood definitions in calculus.

idk1029
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Hi guys,

I just started reading an introductory book on analysis. I'm up to the part where they talk about functions now, and I'm getting lost.

The theorem that I'm having trouble envisioning is: Let f: D-> R and let c be an accumulation point of D. Then limx->cf(x)=L iff for each neighborhood V of L there exists a deleted neighborhood U of c such that f(U\bigcapD) is contained in V.

Why is it N*(c) rather than just N(c)? There's a picture in the book of the deleted point corresponding to L and...I think it's just confusing me more. First theorem in the book that I couldn't wrap my head around. :(
 
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idk1029 said:
The theorem that I'm having trouble envisioning is: Let f: D-> R and let c be an accumulation point of D. Then limx->cf(x)=L iff for each neighborhood V of L there exists a deleted neighborhood U of c such that f(U\bigcapD) is contained in V.

Why is it N*(c) rather than just N(c)?

[STRIKE]Because c \notin D in general. A set is closed iff it contains all of its accumulation points.[/STRIKE]

Edit: Scratch that. Even if c \notin D, then c \notin U \cap D and so it wouldn't be problem.

What I should have said is that f may not be continuous at c.
For example D = (-1,1) and f(0) = 1 but f(0) = 0 elsewhere. We need a definition so that the limit is still 0 at x = 0, even though f(0) is not equal to 0.
If we were using N(c), then when V = (-1/2,1/2), then f(0) \notin V.
 
Last edited:
It seems straight forward that if the limit is L, then there is a neighborhood by definition, and if you delete a point, then it is still a neighborhood.
(Though we may be assuming we're in a certain type of space, I'm not sure which development is in your book, so you may want to check that you've told us everything we need to know.)

So my advice, try to look for the unusual part in the other direction that shows the L is the limit, or supposes it doesn't. If you pull apart that proof, you may gain a foothold in developing the intuition.
 
The basic idea of "limit" is that if \lim_{x\to a} f(x)= L then if x is very, very close to a, the f(x) is very, very close to L. But we are NOT concerned with what happens at x= a. (One important reason for that is that we want to use the limit to find derivatives which limits of "rate of change" calculations, \Delta y/\Delta x which does not exist at the target point.)

Let f(x)= 2x if x is NOT equal to 1 and f(1)= 3. If x is very very close to 1 but not equal to 1, f(x) will be very very close to 2. That is the result we want, not the "3" which just happens to be the value of f(1). To avoid that, we "delete" x= 1 from the neighborhood.
 
Thanks guys! I get it now. You guys are spot on. I was thinking about the point at c and continuity instead of just focusing on the limit. Ha. Now I'm embarrassed. :redface:
 

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