[a, b]

Definition

[a, b] denotes a closed interval in the context of real numbers, including all numbers x such that a ≤ x ≤ b. This means that both endpoints, a and b, are included in the interval. Closed intervals are crucial in mathematical analysis because they allow for the consideration of continuous functions over defined ranges and provide a means to discuss properties such as limits, continuity, and boundedness within those bounds.

5 Must Know Facts For Your Next Test

1. The closed interval [a, b] can also be represented in set notation as {x | a ≤ x ≤ b}.
2. Closed intervals are compact in the real number system, meaning they are closed and bounded.
3. The endpoints of the closed interval [a, b] play an important role in defining continuity and differentiability at those points.
4. In calculus, the Intermediate Value Theorem applies to continuous functions defined on closed intervals.
5. When graphing a closed interval [a, b], solid dots are used at points a and b to indicate that these endpoints are included.

Review Questions

• How does the concept of closed intervals relate to continuity in real-valued functions?
  • Closed intervals are essential for understanding continuity because they provide a complete range over which a function can be examined. A function is considered continuous on a closed interval [a, b] if it does not have any jumps or breaks within that interval. Since both endpoints are included, it ensures that limits from either side of those endpoints coincide with the function's values at those points.
• Explain how closed intervals differ from open intervals and the implications this has for mathematical analysis.
  • Closed intervals [a, b] include both endpoints a and b, while open intervals (a, b) exclude them. This difference affects various properties in mathematical analysis. For instance, functions defined on closed intervals may achieve maximum and minimum values according to the Extreme Value Theorem, which does not necessarily hold for functions defined on open intervals. This distinction is crucial when determining bounds and behaviors of functions.
• Evaluate the significance of closed intervals in terms of compactness and its effects on convergence in sequences.
  • Closed intervals are significant because they are compact sets in the real number system, meaning every open cover has a finite subcover. This property leads to vital conclusions about sequences: any sequence within a closed interval must have a convergent subsequence that converges to a limit within that same interval. Therefore, understanding closed intervals lays the groundwork for more advanced concepts like sequential compactness and convergence properties in real analysis.

Related terms

Open Interval: An open interval (a, b) includes all numbers x such that a < x < b, excluding the endpoints a and b.

Half-Open Interval: A half-open interval [a, b) includes all numbers x such that a ≤ x < b, including a but excluding b.

Real Numbers: Real numbers encompass all rational and irrational numbers, representing points on an infinite line called the number line.