## Interval Definition

The subset of real numbers comprised between any two of its elements is called an interval: a and b; which are called *extremes of the interval*.

Geometrically, the intervals correspond to line segments, rays or the same real line.

## Closed interval

It is the set of real numbers formed by a, b and all the elements between them. That is, **it is that interval whose ends belong to the given interval** and is represented by brackets [ ].

[a,b] = \{ x / a \le x \le b \}

With respect to graphical notation, it is denoted by filled circles, denoted as closed. That is, as shown in the following image:

## Open interval

It is the set of real numbers between a and b. That is to say,** it is that interval whose extremes are not contained in the interval**, they only serve as borders. To represent this range, parentheses ( ) are used.

(a,b) = \{ x / a < x < b \}

With respect to graphical notation, it is denoted by empty circles, denoted as open. That is, as shown in the following graph:

## Semi-open or semi-closed interval

It is that interval that **does not contain one of the extremes**, it can be located to the right or left. It is represented by combining parentheses and brackets: ( ] or [ ). Namely:

Interval half-open to the left (or half-closed to the right)

(a,b] = \{ x / a < x \le b \}

Interval half-open to the right (or half-closed to the left)

[a,b) = \{ x / a \le x < b \}

## Infinite interval

**It is that interval that has at least one of its ends at infinity** \infty. Infinity, being an immeasurable quantity (not measurable), uses parentheses for its representation.

[a,+\infty) = \{x / x \ge a \} | (a,+\infty)=\{ x / x > a \} |

(-\infty, b] = \{ x / x \le b \} | (-\infty,b) = \{x / x < b \} |

(-\infty,+\infty) = \mathbb{R} |

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