1 Definition and basic properties

A subinterval is an interval contained within a larger interval. In elementary calculus, the term usually refers to the set of all points between two selected points of the original interval. Subintervals are used to divide a domain into smaller parts so that functions, sums, and approximations can be studied piece by piece.

The idea is especially important when the original interval is too large or too complex for direct analysis. By working on smaller pieces, one can estimate areas, compare function values, and formulate limit processes more effectively.

1.1 Interval containment

For an interval \(I\), a subinterval is any interval \(J\) such that every point of \(J\) also lies in \(I\). If \(I\) is \([a,b]\), then intervals such as \([c,d]\), \((c,d)\), or \([c,d)\) can serve as subintervals provided \(a \le c \le d \le b\).

Containment is the defining feature. A set may be inside an interval without itself being an interval, but it is not usually called a subinterval unless it has the interval property of containing all points between its endpoints.

1.2 Endpoints of a subinterval

A subinterval is determined by its endpoints, which are the selected boundary points of the smaller interval. These endpoints may or may not belong to the subinterval, depending on the interval type.

In many applications, the endpoints are chosen from a larger interval and used to mark the beginning and end of a segment. The distance between them gives the length of the subinterval, a quantity that plays a central role in approximation and integration.

1.3 Types of subintervals

Subintervals may be open, closed, or half-open, depending on whether endpoints are included. The choice affects how the interval behaves in limit arguments and in definitions involving partitions.

1.3.1 Open subintervals

An open subinterval has the form \((c,d)\). It includes all points strictly between \(c\) and \(d\), but not the endpoints themselves. Open subintervals are often used in theoretical arguments involving interior points and continuity.

1.3.2 Closed subintervals

A closed subinterval has the form \([c,d]\). It includes both endpoints as well as every point between them. Closed subintervals are especially common in analysis because many fundamental theorems are stated for closed and bounded intervals.

1.3.3 Half-open subintervals

A half-open subinterval includes one endpoint but not the other, such as \([c,d)\) or \((c,d]\). Such intervals are useful in partitions, where adjoining pieces should fit together without overlapping at every boundary point.

2 Subintervals in partitions

Partitions divide an interval into a finite collection of subintervals. This decomposition is a standard starting point for Riemann sums, numerical quadrature, and many approximation schemes.

The structure of the partition determines the size and arrangement of the pieces. Smaller subintervals generally lead to better approximations, especially when the function changes rapidly across the interval.

2.1 Partition of an interval

A partition of \([a,b]\) is a finite ordered set of points \(a=x_0<x_1<\cdots<x_n=b\). These points divide the interval into subintervals \([x_{i-1},x_i]\) for \(i=1,\dots,n\).

Each subinterval covers one portion of the original domain, and together they span the whole interval. In this way, a partition turns a single interval into a chain of manageable segments.

2.2 Partition points

Partition points are the selected points that mark the boundaries between adjacent subintervals. They determine both the number of pieces and their sizes.

The placement of these points matters. If they are spread evenly, the subintervals have equal length; if they are clustered in one region, the partition is finer there and coarser elsewhere.

2.3 Length of a subinterval

The length of a subinterval with endpoints \(c\) and \(d\) is \(d-c\). In a partition of \([a,b]\), the lengths of the subintervals indicate how the total interval has been divided.

Length is a key measure in estimation. Many convergence results depend on whether the longest subinterval becomes small as the partition is refined.

2.3.1 Equal-length subintervals

Equal-length subintervals all have the same size. If \([a,b]\) is divided into \(n\) equal parts, each subinterval has length \((b-a)/n\).

These are convenient in computation because they simplify formulas and reduce bookkeeping. They are common in introductory numerical methods and in basic Riemann-sum constructions.

2.3.2 Unequal-length subintervals

Unequal-length subintervals vary in size within the same partition. They may be chosen to place more points where a function changes quickly or where greater accuracy is desired.

Unequal subdivisions are useful in adaptive methods and in applications where uniform spacing is inefficient. They can capture local behavior more accurately than a coarse equal partition.

3 Subintervals in Riemann integration

In Riemann integration, subintervals provide the framework for approximating the area under a curve. The interval of integration is divided into smaller pieces, and function values on those pieces are used to build sums.

This approach connects geometry with limits. As the subintervals shrink, the approximation may converge to the exact integral when the function is integrable.

3.1 Role in Riemann sums

A Riemann sum assigns a rectangle to each subinterval of a partition. The width of each rectangle is the length of the subinterval, and its height is determined by a function value chosen from within that piece.

The sum of these rectangles estimates accumulated quantity, such as area, total mass, or displacement. Finer subintervals usually improve the estimate.

3.2 Sample points within subintervals

For each subinterval, one selects a sample point where the function is evaluated. The sample point may be leftmost, rightmost, middle, or chosen in another prescribed way.

Different sample-point rules lead to different approximations. Although the partitions may be the same, the resulting sums can vary until the subintervals become very small.

3.3 Upper and lower sums

Upper and lower sums use the greatest and least function values on each subinterval, respectively. The upper sum overestimates the integral, while the lower sum underestimates it.

These sums are useful for bounding the true value of the integral. When the two sums approach the same limit under refinement, the function is integrable in the Riemann sense.

4 Subintervals in numerical methods

Subintervals also appear in numerical methods that approximate functions, derivatives, and integrals. Breaking a domain into smaller pieces often makes computation more stable and more accurate.

Many algorithms depend on how the interval is subdivided. The choice of subinterval size can affect speed, precision, and the amount of error in the final result.

4.1 Subdivision for approximation

Subdivision replaces a difficult whole interval with simpler local pieces. On each subinterval, a method may use a constant, linear, or higher-order approximation.

This strategy is common in trapezoidal, Simpson-type, and piecewise interpolation methods. The local approximations are then combined to estimate the overall quantity.

4.2 Error considerations

Approximation error is often related to the length of the subintervals. Smaller pieces typically reduce error because the function is better represented locally.

However, very small subintervals increase computational cost. Practical methods balance accuracy against efficiency by choosing a partition size suited to the problem.

4.3 Adaptive subdivision

Adaptive subdivision adjusts subinterval sizes according to the behavior of the function. Regions with rapid change are divided more finely, while smoother regions are left with larger pieces.

This approach improves efficiency by concentrating work where it is most needed. It is widely used in numerical integration and in algorithms that estimate error dynamically.

Subintervals are closely connected to several basic ideas in mathematics. They are a specific kind of interval, they are subsets of larger sets, and they become especially useful when partitions are refined.

5.1 Interval

An interval is a set of real numbers containing every number between any two of its points. Subintervals inherit this property while occupying only part of a larger interval.

5.2 Subset

A subset is any set whose elements are all contained in another set. A subinterval is a subset with additional structure: it must itself be an interval.

5.3 Partition refinement

Partition refinement means replacing a partition by one with more points and smaller subintervals. This process is central to convergence arguments in integration and approximation.