Definition

1. Noun:
   • A closed interval in mathematics: A "bounded interval" is a specific type of interval on the real number line that includes both of its endpoint numbers. It is defined by the set of all numbers x such that a ≤ x ≤ b, where a and b are the endpoints. The term emphasizes that the interval is finite and has defined lower and upper bounds.

Usage

• The term "bounded interval" is primarily used in mathematical contexts, specifically in calculus, analysis, and set theory, to describe a closed and finite segment of the real number line.
• It is often contrasted with an "open interval" (which does not include endpoints) or an "unbounded interval" (which extends to infinity).

Examples

• Noun:
  • The function is continuous on the bounded interval [0, 5]. (The function's behavior is defined for all numbers from and including 0 to and including 5.)
  • A bounded interval, such as [-1, 1], contains a maximum and a minimum value. (The numbers -1 and 1 are both included and are the smallest and largest values in that set.)

Advanced Usage

• "Bounded interval" vs. "Closed interval": In precise terminology, "bounded interval" explicitly notes the interval's finite extent, while "closed interval" explicitly notes the inclusion of endpoints. The interval is both closed and bounded. However, an interval can be bounded but not closed (e.g., is bounded but open), so the terms are not perfectly synonymous.
• In proofs and theorems: The concept is fundamental. For example, the Extreme Value Theorem states that a continuous function on a must attain a maximum and a minimum value.

Variants and Related Words

• Closed Interval (n): Often used synonymously with "bounded interval" when the endpoints are included. Formally written as .
• Open Interval (n): An interval that does include its endpoints, written as . It is bounded but not closed.
• Interval (n): The general term for a connected set of real numbers.
• Bounded (adj): Having limits or bounds. A set is bounded if all its elements lie within some fixed distance from a point.

Synonyms

• Closed interval: (When context clarifies it is finite.)
• Finite interval: (Emphasizes the length is not infinite.)

Related Phrases

• Compact interval: In topology and advanced analysis, a closed and bounded interval on the real line is called a compact interval, due to the Heine-Borel theorem.

Noun

1. an interval that includes its endpoints