ABSTRACT
The reader should verify that the intersection of finitely many neighborhoods of a is again a neighborhood of a and that neighborhoods separate points, that is, if a 6= b are extended real numbers, then there exist neighborhoods N (a) and N (b) such that N (a) ∩N (b) = ∅. 3.1.2 Definition. An accumulation point of a nonempty set E of real numbers is an extended real number a such that every neighborhood of a contains a point of E not equal to a. A member of E that is not an accumulation point of E is called an isolated point of E. ♦
[−1, 0] ∪ {+∞}, and the set of isolated points of E is N. The following definition of limit is sufficiently general to include the usual
3.1.3 Definition. Let E ⊆ R, let f be a real-valued function whose domain includes E, and let a, L ∈ R, where either a ∈ E or a is an accumulation point of E (not necessarily in the domain of f). We write
L = lim x→a x∈E
f(x)
if, for each neighborhood N (L) of L, there is a neighborhood N (a) of a such that
x ∈ E ∩N (a) implies f(x) ∈ N (L). (3.1) In this case we say that that f(x) approaches L as x tends to a along E ♦
The restrictions on a guarantee that E ∩N (a) 6= ∅, hence condition (3.1) is not vacuously satisfied. Note that if a ∈ E is not an accumulation point of E, then it must be an isolated point, in which case lim{x→a, x∈E} f(x) trivially exists and equals f(a).
