Choosing Partitions For Tighter Estimates

Why Partition Placement Matters

A partition divides an interval into subintervals. On each subinterval, the upper Darboux sum uses the function’s supremum, while the lower Darboux sum uses its infimum. Their difference measures the spread between these two estimates.

Making every subinterval equally wide is a simple way to refine an estimate. But equal widths need not be the best choice when the function varies much more in some regions than in others. A wide interval where the function is nearly constant may contribute little to the gap, while a short interval containing a steep change can contribute more.

The practical principle is to examine each interval’s contribution: its width multiplied by the function’s oscillation there. If the oscillation is large in one region, placing more partition points there can reduce the total gap. This is a guide, not a guarantee: the actual sums must be calculated or bounded.

The Darboux Gap

Let \(f\) be bounded on \([a,b]\), and let \(P\) be a partition \(a=x_0<x_1<\cdots<x_n=b\). On the \(i\)th subinterval \([x_{i-1},x_i]\), define its width as \(\Delta x_i=x_i-x_{i-1}\), its supremum as \(M_i\), and its infimum as \(m_i\). The upper and lower Darboux sums are

Subtracting the sums gives the gap:

Every term in the gap is nonnegative. Therefore, to reduce the gap, a partition should break up intervals whose width and oscillation together make a large contribution. If \(f\) is Riemann integrable, its integral lies between the lower and upper sums; a small gap therefore gives a tight pair of bounds.

Worked Example: More Points Where the Function Changes

Why Equal Widths Can Be Effective

Equal widths are not automatically inferior. If variation is distributed fairly evenly, they can give a good estimate and are easy to manage. A useful general bound applies when \(f\) is Lipschitz on \([a,b]\): this means there is a constant \(K\ge0\) such that \(|f(x)-f(y)|\le K|x-y|\) for all \(x,y\in[a,b]\).

To justify the gap bound, take any two points \(x,y\) in the same subinterval \([x_{i-1},x_i]\). By the Lipschitz condition, \(|f(x)-f(y)|\le K\Delta x_i\). Taking the greatest possible difference on that interval gives \(M_i-m_i\le K\Delta x_i\). Multiplying by the width and summing proves the first inequality:

For the second claim, the widths are positive and sum to \(b-a\). The identity \(\sum_{i=1}^{n}(\Delta x_i-(b-a)/n)^2\ge0\), expanded and simplified, gives

Equality holds exactly when every width is \((b-a)/n\). This proves that equal widths minimize this particular general bound. It does not prove that equal widths minimize the actual Darboux gap for every function. If the function’s variation is concentrated in one region, an uneven partition can do better, as the preceding example showed.

Worked Example: A Monotone Function with Two Subintervals

How Variation Guides the Choice

For each subinterval, the contribution is \((M_i-m_i)\Delta x_i\). When choosing points, inspect both factors. A large oscillation suggests subdividing that region; a large width can also make the contribution large, even if the oscillation per unit of width is moderate. If a region is constant, subdividing it does not reduce the gap there.

When a useful bound on local variation is available, it can guide the placement more precisely. For example, if the function changes at most \(K_i\) times the distance between any two points of a particular subinterval, then that subinterval contributes at most \(K_i(\Delta x_i)^2\). Larger local values of \(K_i\) make smaller widths desirable for controlling this bound. This is a planning tool: the constants may be unavailable, and the bound need not equal the actual gap.

A practical process is:

Worked Example: Uneven Widths Can Make the Gap Worse

A Common Mistake

A common error is to assume that having a smaller interval somewhere guarantees a smaller total gap. For \(f(x)=x\) on \([0,1]\), one might incorrectly compare the widths \(1/2,1/4,1/4\) with the equal widths \(1/3,1/3,1/3\) and conclude that the uneven partition must be better because it contains two shorter intervals. That conclusion ignores the remaining width and the sum of all contributions.

The corrected calculation is

The comparison must include every subinterval. The quantity to reduce is the sum of width times oscillation, not the width of one selected interval or the number of short intervals.

Choosing a Partition in Practice

If little is known about where the function varies, equal widths are a reliable and transparent starting point. If a graph or formula indicates concentrated variation, an uneven partition may produce a smaller gap with the same number of intervals. In either case, the claim is verified by calculating or bounding the contributions on all subintervals.

The key question is not simply, “Are the widths equal?” It is, “Where does this partition’s Darboux gap come from?” Once the large contributions are identified, partition points can be moved toward the regions responsible for them. The resulting estimate is only as good as the comparison supporting it.