Definition

1. Noun:
   • A set of real numbers that includes all numbers between two given endpoints, and also includes the endpoints themselves. In mathematics, a closed interval is a continuous range of numbers. It is defined by its lower bound and its upper bound, and both of these boundary numbers are considered part of the set.

Usage

• The concept of a closed interval is fundamental in calculus, analysis, and set theory.
• It is typically denoted using square brackets. For example, the closed interval from to is written as [a, b].
• This contrasts with an , which does not include its endpoints and is written with parentheses: (a, b).

Examples

• Noun:
  • The function is continuous on the closed interval [0, 5]. (This means the function's behavior is defined and smooth for every number from 0 to 5, including 0 and 5.)
  • All real numbers x satisfying 1 ≤ x ≤ 10 belong to the closed interval [1, 10].
  • Find the maximum value of the function over the closed interval [-2, 3].

Advanced Usage

• "Closed interval method": A technique in calculus for finding the absolute maximum and minimum values of a continuous function on a closed interval [a, b]. The method involves evaluating the function at its critical points at the endpoints and .
• Nested closed intervals: A sequence of closed intervals where each interval is contained within the previous one (e.g., [0, 1], [0.2, 0.8], [0.3, 0.7]...). The Nested Interval Theorem states that the intersection of such intervals contains at least one real number.

Variants and Related Words

• Interval (n): The general term for a set of numbers between two boundaries.
• Open interval (n): An interval that does include its endpoints, denoted (a, b).
• Half-open interval (n): An interval that includes only one of its endpoints, denoted [a, b) or (a, b].
• Bounded interval (n): An interval with finite endpoints, which can be open, closed, or half-open.

Synonyms

• Inclusive interval: (Less common) Emphasizes the inclusion of the endpoints.

Antonyms

• Open interval: An interval that excludes its endpoints.

Noun

1. an interval that includes its endpoints