closed interval

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closed interval

A number line shows a closed interval from 2 to 5.

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.
closed interval

A number line shows a closed interval from 2 to 5.

Noun
  1. an interval that includes its endpoints

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