An
interval is a
set of real numbers with no gaps: whenever it contains two numbers, it contains every real number between them. Its boundaries may be included or excluded, and the range may extend indefinitely in either direction.
In finance, intervals describe admissible rates, allocation constraints and time windows. The endpoint convention answers a practical question: does a value exactly on the boundary qualify?
Reading the brackets
For finite numbers a<b, the four bounded forms are:
| Notation | Membership rule | Name |
|---|
| [a,b] | a≤x≤b | Closed interval: includes both endpoints |
| (a,b) | a<x<b | Open interval: excludes both endpoints |
| [a,b) | a≤x<b | Half-open: includes the left endpoint |
| (a,b] | a<x≤b | Half-open: includes the right endpoint |
A square bracket includes its endpoint; a parenthesis excludes it. These conventions agree with the
University of Queensland's Mathematical Foundations notes, p. 141.
The pair of numbers gives the boundaries,
not a list of the only allowed values. For example,
[2,5] contains
3.5, whereas
{2,5} contains only two points. A quoted range of real numbers also includes irrational values between its boundaries; it is not restricted to integers or displayed decimal ticks.
A rate screen with a strict upper cutoff
Suppose a hypothetical screen accepts an annual rate of
at least 2% but strictly less than 5%. Let
r be the annual rate expressed as a decimal. The eligible interval is
No compounding calculation is needed: this is a rule about the supplied rate input. Any comparison of actual products would first need consistent rate conventions.
| Annual rate | Decimal input | Eligible? |
|---|
| 1.999% | 0.01999 | No: below the lower boundary |
| 2% | 0.02 | Yes: lower boundary included |
| 3.5% | 0.035 | Yes: lies between the boundaries |
| 4.999% | 0.04999 | Yes: still below the upper boundary |
| 5% | 0.05 | No: upper boundary excluded |
Original hypothetical rule. The axis displays annual rates in percent, while interval notation uses decimals. Filled boundary markers mean included; hollow markers mean excluded.
The interval's width is
0.05−0.02=0.03, or
3 percentage points. That is a difference between rates, not a 3% relative increase. Changing an endpoint from included to excluded changes membership, but does not change this width.
Half-open intervals are especially useful for adjoining buckets. If the next bucket is [0.05,0.08), exactly 5% belongs to that bucket alone. Closing both buckets at 5% would count it twice; opening both there would leave it unassigned. This same convention helps assign an event exactly at a time-window boundary to one window.
Unbounded ranges and special cases
A nonnegative input has domain [0,∞); a strictly positive input has domain (0,∞). A ceiling alone can give (−∞,b], meaning x≤b. All real numbers form (−∞,∞).
Always put a parenthesis next to infinity: ∞ and −∞ indicate unbounded directions, not real numbers that could be members. An unbounded interval can still have an included finite endpoint, as [0,∞) does.
When the finite boundaries coincide, [a,a]={a} is a single-point interval. By contrast, (a,a) and the half-open versions with equal boundaries are empty. Write the smaller boundary first for a nonempty interval; reversing the bounds does not specify a range in descending order.
What the notation does—and does not—promise
The permitted set must actually contain every intermediate value. If a trading system allows only whole lots, the sizes 2, 3, 4 and 5 form a discrete set; [2,5] alone would also allow fractional sizes. Similarly, two separated allowed bands generally require a union of intervals, rather than one interval spanning the forbidden gap.
Endpoint inclusion can affect whether an optimum is attained. On the hypothetical eligible band
[0.02,0.05), the objective
f(r)=r has minimum 0.02 but
no maximum: every eligible rate can be increased a little while remaining below 0.05. The number 0.05 is the least upper bound, not an eligible maximiser. On the closed band
[0.02,0.05], both extrema are attained. More generally, a continuous real-valued objective on a finite closed interval attains its minimum and maximum.
For that next step,
Maximum and Minimum Values and the Closed-Interval Method develops the endpoint and critical-point search; the full lesson requires
membership, with a public preview available.
Finally, the word
interval supplies no probability claim. A
confidence interval needs a statistical construction and a coverage interpretation. A policy band such as the rate screen above is simply an eligibility rule; it offers no guarantee about a future realised rate.
Sources
- James Stewart, Single Variable Calculus: Early Transcendentals, 6th edition, Appendix A, “Numbers, Inequalities, and Absolute Values,” subsection “Intervals,” especially Table 1, pp. A3–A4 — finite and infinite interval notation and number-line interpretation.
- University of Queensland, MATH1050 Mathematical Foundations, 2006, §5, p. 141 — open, closed and half-open intervals, and unbounded ranges.
The rate-screen example and figure are original hypothetical illustrations.