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- 690

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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