Accumulation point

One of the fundamental concepts of mathematical analysis is that of limit, and in the case of a function it is to calculate the limit when the nearby points approach a fixed point, which may or may not be in the domain of the function, this point is called accumulation point.

Summary

[ hide ]

  • 1 Definition
  • 2 Other considerations
  • 3 References
  • 4 Bibliographic sources

Definition

A point a of S is called the point of accumulation of the set L, part of S, when in every neighborhood of a there is an infinite number of points of L. [1]

Examples

  • Let be the open interval L = (m, n); S = set of all real numbers. The element m, real number, is the point of accumulation of L, since in the neighborhood (m-ε; m + ε) there are infinity of points of L.
  • Let the set L of positive rational numbers x be such that x 2<3 the number 3 5 is the point of accumulation, since there are infinite positive rational numbers, the square of which is less than the square root of 3.
  • Let L be the set of points x = 2-1 / n, where n is a positive integer, the rational number 2 is the point of accumulation of L.

An accumulation point may or may not belong to the given set. In the examples above, none of the accumulation points is in the case as a whole.

In the case of the open interval (m, n) any point of it is accumulation point.

Other considerations

Isolated point

given the point h of L, this is an isolated point, if it is in L, also in a certain neighborhood there is no other point of L.

Let the set L = (2,9) \ (4,7) ∪ {6}, let be an isolated point of L.

Derived set

Given the set L, the set of all its accumulation points is called the derived set .

Adherence

The set L and all its accumulation points is called the adherence of L, which is denoted Adh L.

The adherence of the open interval (m; n) is the closed interval [m, n]

The set F, part of S, is called the closed set if F is equal to its adherence [2]

Set A, part of S, is called open if its complement S \ A is closed

In analysis, the limit of a function is calculated at an accumulation point of the domain. The limit of f (x) = ln x can be calculated at point 0, which is not in the domain or definition field, but it is the accumulation point of the domain.

 

Leave a Comment