Question

set of limit points of set of all integral points of the real line R is​

Answer 1

Explanation:

The confusion here is that the definition of a limit point x in a set X requires that every neighborhood of x contains a point of X different from x. A limit point is also known as an accumulation point.

This is in contrast to the definition of an adherent point, also known as a contact point, which is a point whose every neighborhood intersects X.

If a set is closed, then every one of its points are adherent points; but not necessarily limit points. For example the set [0,1]∪{2} is a closed set in R. Every point is an adherent point, but 2 is not a limit point.

This terminology a common point of confusion.

To answer the original question, the integers have no limit points in the reals, since all integers are isolated; that is, each integer has a neighborhood that does not contain any other integers.