A bounded interval is a set of real numbers that has both a lower and an upper limit. If you picture the number line, a bounded interval is a segment with a definite starting point and a definite ending point, even if one or both of those endpoints aren’t included in the set itself. The interval from 2 to 7, for instance, is bounded because every number in it sits between two finite values.
The Core Idea
An interval is bounded when there exists some real number smaller than all its elements (a lower bound) and some real number larger than all its elements (an upper bound). If either side stretches off toward infinity, the interval is unbounded. So the set of all numbers greater than 5 is unbounded because it has no upper limit, while the set of numbers between 5 and 10 is bounded because it’s fenced in on both sides.
In more precise terms, a bounded interval takes the general form of all values x between two real numbers a and b, where a is less than b. The key distinction from unbounded intervals is that both a and b are finite numbers, not infinity.
Four Types of Bounded Intervals
Bounded intervals come in four varieties, depending on whether each endpoint is included or excluded. The notation uses square brackets to mean “included” and parentheses to mean “excluded.”
- Open: (a, b) includes all x where a < x < b. Neither endpoint belongs to the set. Example: (2, 5) contains 3 and 4.9 but not 2 or 5.
- Closed: [a, b] includes all x where a ≤ x ≤ b. Both endpoints belong to the set. Example: [2, 5] contains 2, 5, and everything between them.
- Left-closed, right-open: [a, b) includes all x where a ≤ x < b. The left endpoint is in, the right is out. Example: [2, 5) contains 2 but not 5.
- Left-open, right-closed: (a, b] includes all x where a < x ≤ b. The left endpoint is out, the right is in. Example: (2, 5] contains 5 but not 2.
All four types are bounded. The choice between them depends on whether you need the boundary values themselves to be part of the set, which matters in different mathematical contexts.
Set-Builder Notation
You’ll often see bounded intervals written in set-builder notation, which spells out the condition each element must satisfy. The closed interval [1, 3] can be written as {x | 1 ≤ x ≤ 3}, which reads “the set of all x such that x is greater than or equal to 1 and less than or equal to 3.” A half-open interval like [10, 30) becomes {x | 10 ≤ x < 30}.
Both notations describe exactly the same set. Interval notation is more compact, while set-builder notation makes the inequalities explicit. You’ll encounter both in textbooks and coursework, so it helps to move between them comfortably.
How Bounded Differs From Unbounded
Unbounded intervals extend infinitely in at least one direction. The interval (3, ∞) has a lower bound of 3 but no upper bound, so it’s unbounded. The interval (-∞, ∞), which is the entire real number line, is unbounded in both directions. Any time you see an infinity symbol in the notation, the interval is not bounded.
A bounded interval, by contrast, always has a finite length. That length equals b minus a, regardless of whether the endpoints are included. The intervals (2, 5), [2, 5], and [2, 5) all have a length of 3. Including or excluding an endpoint doesn’t change the length because individual points have zero width on the number line.
Supremum, Infimum, and Endpoints
Two related concepts show up frequently alongside bounded intervals: the supremum (least upper bound) and the infimum (greatest lower bound). For a bounded interval between a and b, the infimum is a and the supremum is b, regardless of whether those endpoints are included in the set.
The distinction matters when you ask whether the interval has a maximum or minimum. If the supremum actually belongs to the set, it’s called the maximum. If the infimum belongs to the set, it’s called the minimum. So the closed interval [2, 5] has a minimum of 2 and a maximum of 5. The open interval (2, 5) still has an infimum of 2 and a supremum of 5, but it has no minimum or maximum because neither endpoint is included. You can get arbitrarily close to 2 or 5 inside the interval, but you never reach them.
Why Bounded Intervals Matter in Calculus
Bounded intervals play a central role in several foundational theorems. The Extreme Value Theorem, for example, guarantees that a continuous function on a closed, bounded interval [a, b] reaches both a maximum and a minimum value somewhere on that interval. This guarantee breaks down if the interval is open or unbounded, because the function might approach but never reach its extreme values.
Closed and bounded intervals (called compact sets in more advanced courses) also ensure that continuous functions behave in predictable, well-controlled ways. On a closed, bounded interval, continuity and uniform continuity are equivalent, meaning a continuous function can’t oscillate or change speed in pathological ways. These properties make closed bounded intervals the standard domain for defining integrals, proving convergence results, and analyzing function behavior throughout calculus and real analysis.
A single point, like {3}, is technically a bounded interval where a equals b. Written as [3, 3], it satisfies the definition since it has both a lower and upper bound. This edge case, sometimes called a degenerate interval, has a length of zero but still counts as bounded.