The Cauchy Integrability Criterion

Suppose you are handed a bounded function on a closed interval. You are asked a yes-or-no question: is it Riemann integrable? The definition of the Darboux Integral tells you to compute two numbers. These are the upper and lower integrals. You must check whether they agree. But these are infima and suprema over all partitions. This is an uncountable and unwieldy collection. Checking equality of two limiting quantities directly is often hopeless in practice. What is needed is a workable test. It must replace an existential statement about a limit. In its place, a concrete, checkable condition on a single, cleverly chosen partition is needed. The Cauchy Integrability Criterion supplies exactly this. It says that $f$ is integrable precisely when a condition holds. For any tolerance $\varepsilon > 0$ , you can produce one partition $P$ . For that partition, the upper and lower Darboux sums are within $\varepsilon$ of each other. This transforms an abstract equality of suprema and infima into a hands-on construction problem. You build a partition, estimate the gap $U(f,P) - L(f,P)$ , and make it small. It is the single most useful tool for proving integrability results. It is the bridge between the definition of the integral and every subsequent integrability theorem. The criterion matters for an important reason. Virtually every later theorem about integrability is proved this way. This includes theorems for continuous functions, for monotone functions, and for functions with finitely many discontinuities. Each is proved by exhibiting a partition satisfying this criterion. Without it, one would be forced to argue directly with suprema and infima. These belong to the upper and lower integrals, and such argument is far more delicate. With the criterion, the argument becomes constructive. You partition the domain adaptively, control oscillation on each piece, and sum up controlled errors weighted by subinterval lengths. This article states the criterion precisely. It proves both directions in full, using only the definition of the Darboux integral. It also uses the refinement lemmas that precede it. It illustrates the technique on two genuinely different functions. It closes by previewing how the criterion becomes the engine behind the integrability of continuous and monotone functions.

Key takeaways

The criterion converts "upper integral equals lower integral" into a checkable condition on a single partition.
$f$ bounded on $[a,b]$ is integrable $\iff$ $\forall \varepsilon>0\ \exists P$ with $U(f,P)-L(f,P)<\varepsilon$ .
The hard direction uses refinement: a common refinement of two "good" partitions controls both sums simultaneously.
The easy direction is a direct squeeze using the definitions of $\underline{\int} f$ and $\overline{\int} f$ .
Boundedness of $f$ is essential; without it, $U(f,P)$ or $L(f,P)$ may not even be finite.
Key idea: integrability is equivalent to the ability to trap $f$ between step functions of arbitrarily small total discrepancy.

Where this lives today

The Cauchy Integrability Criterion is the workhorse behind essentially every classical integrability theorem taught after it: it is used verbatim to prove Integrability of Continuous Functions (using uniform continuity to control oscillation on small subintervals), Integrability of Monotone Functions (using monotonicity to bound the total oscillation by a telescoping sum), and the integrability of functions with finitely many discontinuities. It underlies the proof of the Equivalence of Riemann and Darboux Integrals, connecting the tagged-sum definition of the integral to the supremum/infimum formulation used here. Beyond pure analysis, the same "trap between upper and lower approximations" idea reappears in measure theory (upper and lower Lebesgue sums, and simple function approximation), in numerical analysis (adaptive quadrature methods refine partitions exactly where the local oscillation $M_i - m_i$ is large, mirroring the proof's construction), in computer-assisted proofs of integral bounds (interval arithmetic tracks $M_i, m_i$ rigorously), and in the theoretical justification of Monte Carlo and Riemann-sum-based numerical integration schemes in computational physics and finance, where controlling the discretization error is precisely the problem of bounding $U(f,P)-L(f,P)$ .

:::quiz[Why must f be bounded on [a,b] for the criterion to even make sense?]

Bounded functions are automatically continuous
Without boundedness, M_i or m_i on some subinterval could be infinite, making U(f,P) or L(f,P) undefined as a real number ✓
Boundedness guarantees the partition has finitely many points
Unbounded functions always fail to be integrable regardless of any criterion

Ask the tutor: How does the Cauchy criterion adapt to prove that a bounded function with only finitely many discontinuities is integrable?

f:[a,b]\to\mathbb{R}

Intuition

Think of $U(f,P)$ as the total area of thin rectangles that sit on top of the graph of $f$ on each subinterval, and $L(f,P)$ as the area of rectangles sitting underneath it. The true area, if it exists, is squeezed between these two numbers for every partition. Refining a partition can only push $U$ down and pull $L$ up (this is the content of Refinement and Monotonicity of Darboux Sums), so the gap $U(f,P) - L(f,P)$ measures how much "slack" remains between the upper and lower step-function approximations of $f$ . The criterion says integrability is exactly the statement that this slack can be made arbitrarily small by choosing $P$ well — not that it vanishes for some fixed sequence of partitions automatically, but that for any tolerance you demand, a suitable partition exists. Geometrically: you can always sandwich the graph of $f$ between an upper staircase and a lower staircase whose total area difference is as small as you like. The picture below shows a single subinterval $[x_{i-1},x_i]$ on which $f$ oscillates. The shaded strip between the upper rectangle (height $M_i$ ) and the lower rectangle (height $m_i$ ) has area $(M_i-m_i)\Delta x_i$ ; summing these strips over all subintervals gives exactly $U(f,P)-L(f,P)$ .

Figure

The gap

U(f,P)-L(f,P)

as a sum of shaded oscillation strips

Theorem

f:[a,b]\to\mathbb{R}

f:[a,b]\to\mathbb{R}

f

Proof technique used: direct squeeze argument via infimum/supremum characterizations (( $\Leftarrow$ )), and common-refinement construction combined with the $\varepsilon/2$ -trick (( $\Rightarrow$ )). Neither direction invokes any theorem about integrability itself — only the definitions of $\overline{\int}, \underline{\int}$ and the refinement monotonicity lemma, so the argument is non-circular.

Worked examples

Example 1: $f(x) = x^2$ on $[0,2]$ , using a non-uniform partition. We verify the criterion directly (rather than invoking continuity results) to see the mechanism at work. Let $\varepsilon>0$ . Take the partition

P = \{0,\ 0.5,\ 1.2,\ 2\}

with subintervals $[0,0.5]$ , $[0.5,1.2]$ , $[1.2,2]$ of lengths $\Delta x_1=0.5,\ \Delta x_2=0.7,\ \Delta x_3=0.8$ . Since $f(x)=x^2$ is increasing on $[0,2]$ , on each subinterval $[x_{i-1},x_i]$ we have $M_i = x_i^2$ and $m_i = x_{i-1}^2$ . Compute:

Subinterval 1: $M_1 - m_1 = 0.5^2 - 0^2 = 0.25$ .
Subinterval 2: $M_2 - m_2 = 1.2^2 - 0.5^2 = 1.44 - 0.25 = 1.19$ .
Subinterval 3: $M_3 - m_3 = 2^2 - 1.2^2 = 4 - 1.44 = 2.56$ .

Then

U(f,P)-L(f,P) = \sum_i (M_i-m_i)\Delta x_i = 0.25(0.5) + 1.19(0.7) + 2.56(0.8) = 0.125+0.833+2.048 = 3.006.

This particular $P$ gives a gap of about $3.006$ — not small. The lesson: an arbitrary partition need not work; we must choose $P$ adaptively to force the gap below $\varepsilon$ . Let's redo this properly with a partition into $n$ equal subintervals of $[0,2]$ , each of length $2/n$ , with nodes $x_i = 2i/n$ . Since $f$ is increasing,

M_i - m_i = x_i^2 - x_{i-1}^2 = (x_i-x_{i-1})(x_i+x_{i-1}) = \frac{2}{n}\left(\frac{2i}{n}+\frac{2(i-1)}{n}\right) = \frac{4(2i-1)}{n^2}.

U(f,P)-L(f,P) = \sum_{i=1}^n (M_i-m_i)\cdot \frac{2}{n} = \frac{8}{n^3}\sum_{i=1}^n (2i-1) = \frac{8}{n^3}\cdot n^2 = \frac{8}{n},

using $\sum_{i=1}^n (2i-1) = n^2$ . Given $\varepsilon>0$ , choose any integer $n > 8/\varepsilon$ (possible by the Archimedean property). Then $U(f,P)-L(f,P) = 8/n < \varepsilon$ . This exhibits, for every $\varepsilon$ , a concrete partition satisfying the Cauchy criterion, so $f(x)=x^2$ is integrable on $[0,2]$ — without yet invoking any general theorem about continuous functions.

Example 2: a function with one interior jump discontinuity, non-symmetric heights. Let

f(x) = \begin{cases} 1 & 0 \le x < 1.3 \\ 5 & 1.3 \le x \le 3 \end{cases}

on $[0,3]$ . We show $f$ is integrable via the Cauchy criterion. Let $\varepsilon > 0$ . The only discontinuity is at $c=1.3$ , and $f$ is bounded ( $1\le f \le 5$ ). Choose $\delta>0$ small enough that $0 < \delta < \min(1.3,\, 1.7,\, \varepsilon/8)$ (we will see why $\varepsilon/8$ suffices), and consider the partition

P = \{0,\ 1.3-\delta,\ 1.3+\delta,\ 3\}.

This is a valid partition since $0 < 1.3-\delta < 1.3+\delta < 3$ given the constraint on $\delta$ . Compute $M_i, m_i$ on each piece:

On $[0, 1.3-\delta]$ : $f\equiv 1$ , so $M_1=m_1=1$ , contributing $0$ to $U-L$ .
On $[1.3-\delta,\ 1.3+\delta]$ (length $2\delta$ ): $f$ takes both values $1$ and $5$ here (since the jump at $1.3$ lies in this subinterval), so $M_2=5,\ m_2=1$ , giving $(M_2-m_2)\cdot 2\delta = 4\cdot 2\delta = 8\delta$ .
On $[1.3+\delta,\ 3]$ : $f\equiv 5$ , so $M_3=m_3=5$ , contributing $0$ .

Hence

U(f,P)-L(f,P) = 0 + 8\delta + 0 = 8\delta.

Since we chose $\delta < \varepsilon/8$ , we get $U(f,P)-L(f,P) = 8\delta < 8\cdot(\varepsilon/8) = \varepsilon$ . As $\varepsilon>0$ was arbitrary and in each case we exhibited a valid partition, the Cauchy criterion holds, so $f$ is integrable on $[0,3]$ . (One can further compute $\int_0^3 f = 1(1.3) + 5(1.7) = 1.3+8.5=9.8$ , but the criterion only requires establishing integrability, not the value.) This example foreshadows the general theorem that bounded functions with finitely many discontinuities are integrable: isolate each discontinuity inside a thin subinterval of total length controlled by $\varepsilon$ , and use boundedness to control the oscillation there.

f

Scope, edge cases & common traps

Situation	What happens
$f$ unbounded on $[a,b]$	$U(f,P)$ or $L(f,P)$ can be $\pm\infty$ ; criterion inapplicable (see counterexample)
$f$ bounded but wildly discontinuous (e.g. Dirichlet function)	Criterion correctly detects non-integrability: $U(f,P)-L(f,P)$ stays bounded away from $0$ for every $P$
Only checking one specific partition, not "for every $\varepsilon$ some $P$ "	Invalid use — a single partition with small gap says nothing; must work for all $\varepsilon$
Confusing "exists $P$ for each $\varepsilon$ " with "one $P$ works for all $\varepsilon$ "	These are different statements; the criterion is the former (quantifier order matters)
Using mesh $\to 0$ without linking to a specific $\varepsilon$	Common informal shortcut; must be made precise via the Riemann-sum formulation, a related but distinct equivalence

Common trapThe classic quantifier error

A very common mistake is to state the criterion as: "there is a partition $P$ such that for all $\varepsilon>0$ , $U(f,P)-L(f,P)<\varepsilon$ ." This reverses the quantifiers. The correct statement is $\forall \varepsilon\,\exists P$ , not $\exists P\,\forall \varepsilon$ . The latter would force $U(f,P)-L(f,P) \le 0$ for a fixed $P$ , which combined with $L(f,P)\le U(f,P)$ forces $U(f,P)=L(f,P)$ for that single partition — an absurdly strong and usually false requirement (it would mean $f$ is constant on each subinterval of that one partition). The partition $P$ is allowed, indeed expected, to depend on $\varepsilon$ .

The mathematician behind it

Origins · 1 minAugustin-Louis Cauchy and how this was discovered

Augustin-Louis Cauchy

1789–1857 · Paris, École Polytechnique and Sorbonne

Cauchy was one of the principal architects of rigorous analysis in the nineteenth century, responsible for the $\varepsilon$ -based definitions of limit, continuity, and convergence that underlie modern calculus. Although the specific Darboux formulation of the integral (upper and lower sums via suprema and infima on subintervals) was developed later by Gaston Darboux and Jean Gaston Darboux's contemporaries as a cleaner alternative to Riemann's original definition via tagged sums, the underlying epsilon-delta philosophy — reducing existence and equality statements to explicit, quantitative, checkable inequalities — is Cauchy's signature contribution to analysis, and the criterion bearing his name is a direct descendant of his general "Cauchy criterion" for convergence.

History

- 1821 — Cauchy's Cours d'Analyse establishes rigorous limit-based foundations for continuity and convergence, introducing the general Cauchy-sequence philosophy later adapted to integration.

- 1823 — Cauchy defines the definite integral for continuous functions via sums over partitions, an ancestor of the Riemann sum approach.

- 1854 — Bernhard Riemann's habilitation introduces the Riemann integral via tagged partition sums, extending integrability beyond continuous functions.

- 1875 — Gaston Darboux reformulates Riemann's integral using upper and lower sums (suprema and infima on subintervals), yielding the cleaner Darboux integral used in this article and proving it equivalent to Riemann's definition.

- 1881 — Darboux's work becomes the standard textbook approach to teaching the Riemann integral, cementing the "Cauchy-style"

\varepsilon

–partition criterion as the standard tool for proving integrability.

Is the Cauchy criterion the same as saying the integral exists as a limit of Riemann sums?

Not literally the same statement, but they are provably equivalent. The Cauchy criterion is phrased purely in terms of Darboux upper and lower sums and partitions, with no reference to a limiting process or tagged points. The Riemann-sum formulation says the integral exists if Riemann sums converge to a single number as the mesh of the partition tends to zero, regardless of tag choice. The Equivalence of Riemann and Darboux Integrals theorem shows these two definitions of integrability coincide for bounded functions. In practice, the Cauchy criterion is usually easier to verify directly because it requires exhibiting just one partition per $\varepsilon$ , not controlling all possible tag choices.

Does the criterion tell you the value of the integral?

No. The criterion is purely an existence test: it certifies that $\overline{\int_a^b} f = \underline{\int_a^b} f$ , without computing this common value. To find the actual value of $\int_a^b f$ , you typically need additional tools, such as the Fundamental Theorem of Calculus, direct computation of $\lim U(f,P)$ along a specific sequence of partitions, or known closed forms. In Example 1, we used the criterion only to establish integrability of $x^2$ ; computing $\int_0^2 x^2\,dx = 8/3$ requires a separate limiting computation of the Riemann sums themselves.

Why does the proof need the ε/2 trick instead of just using ε directly?

The two partitions $P_1$ and $P_2$ are found independently, one controlling how close $U(f,P_1)$ is to the upper integral, and the other how close $L(f,P_2)$ is to the lower integral. If each were only guaranteed to be within $\varepsilon$ (not $\varepsilon/2$ ), then after passing to the common refinement, the total gap $U(f,P)-L(f,P)$ could be as large as $2\varepsilon$ , not $\varepsilon$ . Splitting the tolerance in half for each piece ensures that when the two approximations are combined (added together after subtraction), the total error budget is exactly $\varepsilon$ , not $2\varepsilon$ . This is a standard bookkeeping device in analysis proofs.

Can the same partition work for every ε in the criterion?

Generally no, and this is the crux of the quantifier warning above. As $\varepsilon$ shrinks toward $0$ , you typically need finer and finer partitions, because forcing $U(f,P)-L(f,P)$ below a tinier tolerance usually requires more subintervals (as seen in Example 1, where the gap $8/n$ requires $n\to\infty$ as $\varepsilon\to 0$ ). There exist degenerate cases — e.g., $f$ constant — where a single trivial partition (even $P=\{a,b\}$ ) achieves gap $0 < \varepsilon$ for every $\varepsilon$ simultaneously, but this is the exception, not the rule guaranteed by the theorem.

How is this criterion used to prove continuous functions are integrable?

The strategy is to use uniform continuity of $f$ on the compact interval $[a,b]$ (a consequence of the Heine–Borel theorem) to choose, for any $\varepsilon>0$ , a $\delta>0$ such that $|f(x)-f(y)|<\varepsilon/(b-a)$ whenever $|x-y|<\delta$ . Then any partition $P$ with mesh less than $\delta$ satisfies $M_i - m_i \le \varepsilon/(b-a)$ on every subinterval, so $U(f,P)-L(f,P) \le \sum_i \frac{\varepsilon}{b-a}\Delta x_i = \varepsilon$ . This directly verifies the hypothesis of the Cauchy criterion, which then hands you integrability for free. This is the blueprint for Integrability of Continuous Functions.

What role does the refinement lemma play, and could the criterion be proved without it?

The refinement lemma (monotonicity of $U$ and $L$ under refining a partition) is used critically in the necessity direction, specifically to justify that the common refinement $P=P_1\cup P_2$ satisfies $U(f,P)\le U(f,P_1)$ and $L(f,P)\ge L(f,P_2)$ simultaneously. Without this lemma there would be no guarantee that combining two separately-chosen partitions preserves both approximation qualities at once; you cannot simply average or juxtapose $P_1$ and $P_2$ without the monotonicity guarantee. This is why the prerequisite Refinement and Monotonicity of Darboux Sums is essential scaffolding, not an optional aside.

Does the criterion generalize to unbounded intervals or improper integrals?

Not directly. The Darboux framework, and hence this criterion, is built for bounded functions on closed bounded (compact) intervals $[a,b]$ ; both $U(f,P)$ and $L(f,P)$ rely on suprema/infima being finite real numbers on each subinterval. For improper integrals (unbounded intervals or unbounded functions), one instead defines the integral as a limit of proper Riemann/Darboux integrals over an exhausting sequence of bounded subintervals, and separately investigates convergence of that limit. The Cauchy criterion as stated here is a foundational tool strictly within the compact, bounded setting; improper integrals require a distinct, though related, limiting framework built on top of it.