Confused about continuity and limits

In summary, the theorem states that if f is a function from a closed set D into a closed set R, then limx->cf(x) is also a function from V into R. The limit is achieved by deleting a point from the neighborhood V of the accumulation point c. If c is not in D, then f(U\bigcapD) is not in V and the limit does not exist.
  • #1
idk1029
2
0
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[itex]\bigcap[/itex]D) 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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  • #2
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[itex]\bigcap[/itex]D) is contained in V.

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

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

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

What I should have said is that f may not be continuous at c.
For example [itex]D = (-1,1)[/itex] 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 [itex]f(0) \notin V[/itex].
 
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  • #3
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.
 
  • #4
The basic idea of "limit" is that if [itex]\lim_{x\to a} f(x)= L[/itex] 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, [itex]\Delta y/\Delta x[/itex] 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.
 
  • #5
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:
 

Related to Confused about continuity and limits

1. What is the definition of continuity?

The definition of continuity is that a function is continuous at a point if the limit of the function as x approaches that point is equal to the value of the function at that point.

2. How do you determine if a function is continuous?

A function is continuous if it meets the following three criteria: 1) The function is defined at the point in question, 2) The limit of the function as x approaches the point exists, and 3) The limit of the function as x approaches the point is equal to the value of the function at that point.

3. What is a limit in calculus?

In calculus, a limit is the value that a function approaches as its input approaches a certain point. It is used to describe the behavior of a function near a specific point or to determine the behavior of a function at infinity.

4. How do you find the limit of a function?

To find the limit of a function, you can use the limit laws, which include direct substitution, factoring, and rationalizing. You can also use the graph of the function or a table of values to estimate the limit.

5. What is the relationship between continuity and limits?

The relationship between continuity and limits is that a function is continuous at a point if and only if the limit of the function as x approaches that point exists and is equal to the value of the function at that point. In other words, continuity is dependent on the existence and equality of the limit at a specific point.

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