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Continuity of Functions

Continuity of Functions

Appendix: Intervals and Interval Notation

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 axb. The values a and b are included in this interval.
  • The interval [a, b) is the set of all real numbers x with ax < 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 < xb. 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 xa. 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 ax. 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.

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