High School Question about a limit definition

[BWV]

From Rosenlicht, Introduction to Analysis:

Definition: Let E, E′ be metric spaces, let p0 be a cluster point of E, and let f(complement(p0)) be a function. A point q ∈ E" is called a limit of f at p0 if, given any e > 0, there exists a δ > 0 such that if p ∈ E , p < > p0 and d( p, p0) < δ, then d′( f( p), q) < e.

A couple of questions, i get the standard epsilon delta limit definition as you would see it in a Calc I textbook, but the other stuff is confusing me.- what is the significance, if any, of the E’, E’’? I read E -> E’ as just showing the range and domain of the function - but what is E’’ - it seems to come out of nowhere?

- a cluster point in Euclidian space (that is all the book is concerned with) is an arbitrary ball around a limit point, so it already contains the limit point -which is outside the domain of the function- but the points around the limit point that comprise the cluster point are in the domain.

- Is q ∈ p0?

Thinking of how this would apply to a simple function like f(p) =1/p

 

[Mark44]

From Rosenlicht, Introduction to Analysis:

Definition: Let E, E′ be metric spaces, let p0 be a cluster point of E, and let f(complement(p0)) be a function. A point q ∈ E" is called a limit of f at p0 if, given any e > 0, there exists a δ > 0 such that if p ∈ E , p < > p0 and d( p, p0) < δ, then d′( f( p), q) < e.

A couple of questions, i get the standard epsilon delta limit definition as you would see it in a Calc I textbook, but the other stuff is confusing me.- what is the significance, if any, of the E’, E’’? I read E -> E’ as just showing the range and domain of the function - but what is E’’ - it seems to come out of nowhere?

- a cluster point in Euclidian space (that is all the book is concerned with) is an arbitrary ball around a limit point, so it already contains the limit point -which is outside the domain of the function- but the points around the limit point that comprise the cluster point are in the domain.

- Is q ∈ p0?

Thinking of how this would apply to a simple function like f(p) =1/p

I can't read some of what you wrote in your Definition section

1. let f(complement(p0)) be a function. -- In IE, I see a small dashed rectangle just to the left of f. Inside the rectangle I think it says OBJ, but I can't tell for sure.
2. if, given any e > 0 -- The rectangle is just before e > 0.
3. such that if p ∈ E , -- here the rectangle shows up right after E.
4. then d′( f( p), q) < e -- The rectangle is just before e.

These don't show up at all in Google Chrome, so I don't know what the rectangles with OBJ are supposed to mean.

 

[Mark44]

- a cluster point in Euclidian space (that is all the book is concerned with) is an arbitrary ball around a limit point

No, this isn't right. A cluster point isn't a ball -- it is a single point. A ball around a point x consists of all points within the set in question that are within a certain distance from x. The difference between a limit point and a cluster point of a set X is that the limit point doesn't necessarily have to belong to X. The term "accumulation point" is also used in place of "cluster point."

 

[BWV]

Ok, copied and pasted on my tablet which was probably the source of the bad characters. Thanks - ok see where I misunderstood the cluster point - that is just the point in the domain that as the function approaches the limit in the range is approached.

 

[member 587159]

Ok, copied and pasted on my tablet which was probably the source of the bad characters. Thanks - ok see where I misunderstood the cluster point - that is just the point in the domain that as the function approaches the limit in the range is approached.

A cluster point must not be part of the domain of the function.

Consider the function $f: \mathbb{R}_0 \to \mathbb{R}: x \mapsto x/x = 1$

Then $\lim_{x \to 0} f(x)$ exists, while $x=0$ is a point that is not in the domain of the function. It is necessary though that it is a cluster point for the definition to make sense.