An interval on the real line is the set of all numbers that fall between two specified endpoints.
Let a and b be real numbers with a < b. We can have the following types of finite intervals:
- The open interval (a, b) is the set of all real numbers that fall strictly in between a and b. That is, all real numbers x with a < x < b. The values a and b are not included in this interval.
- The closed interval [a, b] is the set of all real numbers x that fall (non-strictly) in between a and b. That is, all real numbers x with a ≤ x ≤ b. The values a and b are included in this interval.
- The interval [a, b) is the set of all real numbers x with a ≤ x < b. a is included in this interval, while b is not.
- The interval (a, b] is the set of all real numbers x with a < x ≤ b. We include b in this interval, but not a.
The values a and b are called endpoints because they're the points at either end of the interval.
To remember which of the two intervals—(a, b) or [a, b]—includes the endpoints a and b, try thinking of the interval notation like arms. If the arms are like the brackets [a,b] then they are holding a and b firmly in there. If arms are like the parentheses (a,b), then the endpoints a and b slip out.
The interval (a, b) is called open, while the interval [a, b] is called closed. The intervals [a, b) and (a, b] are neither open nor closed. We might hear these intervals called "half-closed," "semi-closed," "half-open", or "semi-open".
We can also have infinite intervals. Since ∞ isn't a number, we'll always have parentheses around ∞ or -∞, not closed brackets. Here are the types of infinite intervals we can have, assuming that a is some finite number:
- (-∞, ∞), which is the whole real line. In other words, this is the set of all real numbers.
- (-∞, a) is the set of real numbers x with x < a. This interval does not include a.
- (-∞, a] is the set of real numbers x with x ≤ a. This interval does include a.
- (a, ∞) is the set of real numbers x with a < x. This interval does not include a.
- [a, ∞) is the set of real numbers x with a ≤ x. This interval does include a.
The infinite intervals (-∞, ∞), (-∞, a], and [a, ∞) are closed intervals. The infinite intervals (-∞, a) and (a, ∞) are open.
The infinite intervals (-∞, ∞), (-∞, a], and [a, ∞) are closed intervals. The infinite intervals (-∞, a) and (a, ∞) are open.