Partitions, Darboux Sums, and Integrability

An integral sums the values of a bounded function over an interval. Informally it is the area under the curve, and nothing more; that this area also computes antiderivatives is a separate theorem, proved in the fundamental theorem of calculus. The definition here follows Darboux: approximate the area from below and from above by rectangles, and declare the function integrable when the two approximations agree.¹ Throughout, $f:[a,b]\to \mathbb{R}$ is bounded — unbounded functions are handled separately as improper integrals.

Partitions and Darboux sums

A partition chops $[a,b]$ into finitely many closed subintervals. On each piece, the boundedness of $f$ guarantees a finite infimum and supremum, and those become the heights of an under- and an over-estimating rectangle.

The lower sum is the area of the inscribed rectangles that fit under the graph; the upper sum is the area of the circumscribed rectangles that contain it.

Because $m\le m_i\le M_i\le M$ whenever $m\le f\le M$, and the widths sum to $b-a$, every lower sum sits below every upper sum on the same partition, and both lie inside a fixed band.

In particular the sets of lower and upper sums are bounded, so their supremum and infimum exist.

Lower and upper integrals

The best under-estimate is the largest lower sum; the best over-estimate is the smallest upper sum.

Both numbers exist for any bounded $f$, and need not be equal. The gap between them measures how far the rectangles are from agreeing on a single area.

No partition closes the gap: no matter how fine, the under-estimate is $0$ and the over-estimate is $1$. This function is the standard example of a non-integrable bounded function, revisited in which functions are integrable.

Refinement

To shrink the gap between $L$ and $U$, cut the partition into finer pieces. Adding points can only raise a lower sum and lower an upper sum.

When a subinterval is split, the infimum over each smaller piece is at least the infimum over the whole, so the inscribed rectangles grow; symmetrically the circumscribed rectangles shrink.

One consequence is used repeatedly: any lower sum is below any upper sum, even on unrelated partitions. Given $P_1$ and $P_2$, their common refinement $\tilde{P}=P_1\cup P_2$ satisfies

Taking the supremum over $P_1$ and the infimum over $P_2$ separates the two Darboux integrals in the right order.

The Riemann integral

When the best under- and over-estimates coincide, that common value is the integral.

By definition a Riemann integrable function is bounded, and its integral is squeezed by any global bounds: $m(b-a)\le {\int }_a^{b}f\le M(b-a)$.

The Cauchy criterion for integrability

Checking that a supremum equals an infimum is awkward directly. The following restatement is used in every integrability proof: it requires only one partition on which the two sums are close.

The difference $U(P,f)-L(P,f)={\sum }_i(M_i-m_i)\Delta x_i$ is the total area of the unshaded caps in the Darboux figure, and integrability amounts to driving that total below any $\epsilon$.

Worked examples

Tagged Riemann sums and the equivalence

The Darboux definition is technically convenient, but the original Riemann definition evaluates $f$ at an arbitrary sample point in each subinterval.

Since $m_i\le f(z_i)\le M_i$ on each piece, every tagged sum is squeezed between the lower and upper sums of the same partition:

Both definitions describe the same class of functions and the same number. If $U(P,f)-L(P,f)<\epsilon$, then the tagged sums are trapped in an interval of length $\epsilon$ around the Darboux value, so the mesh-based limit exists and equals it; conversely a convergent tagged limit yields, by choosing tags near the infimum and supremum, matching lower and upper integrals.⁵ The equivalence is why Riemann integral and Darboux integral name the same object.