Closed Interval - English Definition & Meaning

Definition

A closed interval is a range of numbers that includes its endpoints. 🔒 Think of it like a fenced-in area where the fence posts are part of the area. It's represented using square brackets, such as [a, b], meaning all numbers between 'a' and 'b', including 'a' and 'b'. Unlike an open interval, a closed interval contains its boundaries. Closed intervals are crucial in calculus and analysis. They ensure certain theorems and properties hold. They are vital in defining the domain and range of functions.

Etymology

The term 'interval' comes from the Latin word 'intervallum', meaning 'space between ramparts' or 'distance'. 'Closed' indicates that the endpoints are included, sealing off the range. This concept arose with the formalization of real analysis in the 19th century. Mathematicians needed precise ways to define sets of numbers.

Related Words

Examples

• "The closed interval [0, 1] includes 0 and 1, as well as all numbers in between."
• "The function f(x) = √x is defined on the closed interval [0, ∞)."
• "The solution to the inequality x² ≤ 4 is the closed interval [-2, 2]."
• "In optimization problems, constraints are often defined using closed intervals."

Encouragement

Understanding closed intervals is essential for working with inequalities and functions. Practice identifying them on the number line, and you’ll be mastering real analysis in no time! 👍

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