# Question about a limit definition

• B
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 text book, 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
Mentor
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 text book, 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
Mentor
- 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."

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.

Math_QED
Homework Helper
2019 Award
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.

BWV