Question about a limit definition

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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
 

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  • #2
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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.
 
  • #3
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- 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."
 
  • #4
BWV
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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.
 
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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.
 
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