The Riemann integral is the first rigorous construction of the [integral](/page/Integral) — the formalisation of the idea that the area under a curve can be captured as the [limit](/page/Limit) of a sum. It applies to bounded [functions](/page/Function) on closed intervals $[a, b] \subset \mathbb{R}$ and works by trapping the function between upper and lower approximations built from rectangles. When these approximations can be made arbitrarily close, the common value they converge to is the integral. The Riemann integral handles continuous functions, monotone functions, and more generally all bounded functions whose set of discontinuities is small in a precise measure-theoretic sense, but it breaks down for highly irregular functions and does not interact well with pointwise limits of [sequences](/page/Sequence) — limitations that eventually led to the Lebesgue theory.

## Motivation

[motivation] ### The Naive Approach: Counting Squares The intuitive notion of "area under a curve" is ancient, but making it precise is harder than it appears. Given a non-negative function $f: [a, b] \to \mathbb{R}$, one might try to define the area of the region below the graph by overlaying a grid of small squares and counting how many fall inside. This approach has two fatal defects. First, grid squares that straddle the [boundary](/page/Boundary) of the region are ambiguous — they are partly inside and partly outside, and the finer the grid, the more boundary squares there are relative to area gained. Second, this method gives no algebraic handle on the area: there is no formula relating the grid count to the function $f$ itself, so computing the area of even simple regions (a triangle, a parabolic segment) requires a separate geometric argument each time. What is needed is a definition that connects the area directly to the values of $f$ and that comes with a built-in approximation scheme. ### The Antiderivative Approach A second natural idea is to bypass area altogether and define the integral through antiderivatives: declare $\int_a^b f(x)\, dx := F(b) - F(a)$ where $F' = f$. This is computationally powerful — it reduces integration to the problem of finding a primitive — but it has serious conceptual and technical deficiencies. Not every bounded function has an antiderivative; the function $\mathbb{1}_{[0,1]}$ has no differentiable primitive whose derivative equals $\mathbb{1}_{[0,1]}$ at the endpoints. Even when antiderivatives exist, they are not unique (any two differ by a constant) and there is no guarantee that an antiderivative can be found in closed form. More fundamentally, taking $F(b) - F(a)$ as the *definition* of the integral makes the [Fundamental Theorem of Calculus](/theorems/632) a tautology rather than a deep theorem connecting two independently meaningful operations. The integral should have its own definition — rooted in approximation and area — so that its relationship to [differentiation](/page/Derivative) becomes a genuine mathematical discovery. ### Riemann's Insight: Approximation by Rectangles Riemann's resolution is to approximate the area from above and below simultaneously. Partition $[a, b]$ into subintervals and construct two families of rectangles: tall ones that overshoot $f$ on each subinterval (using the supremum of $f$) and short ones that undershoot it (using the infimum). The overshoot gives an upper bound for the area, the undershoot gives a lower bound, and the true area — if it exists — is trapped between the two. As the partition is refined, both families of rectangles fit the curve more tightly. If the upper and lower bounds can be driven together to a common limit, that limit is the integral of $f$. This definition is intrinsic to $f$: it requires no antiderivative, it works for any bounded function for which the upper and lower bounds converge, and it reduces the existence of the integral to the single quantitative condition that the oscillation of $f$ on small intervals is controllable. [/motivation]

## Definition

The construction proceeds in three stages: first the combinatorial notion of a partition, then the upper and lower sums that approximate the integral, and finally the passage to the [supremum and infimum](/page/Supremum%20and%20Infimum) over all partitions.

[definition:Partition] Let $a, b \in \mathbb{R}$ with $a < b$. A **partition** of $[a, b]$ is a finite set $P = \{x_0, x_1, \ldots, x_n\}$ satisfying \begin{align*} a = x_0 < x_1 < x_2 < \cdots < x_n = b. \end{align*} The **mesh** (or **norm**) of $P$ is \begin{align*} \|P\| := \max_{1 \le i \le n} (x_i - x_{i-1}). \end{align*} A partition $Q$ is a **refinement** of $P$ if $P \subseteq Q$ as [sets](/page/Set). [/definition]

Every partition decomposes $[a, b]$ into $n$ non-overlapping subintervals $[x_{i-1}, x_i]$ of widths $\Delta x_i := x_i - x_{i-1}$. Refinement only adds points, so it splits existing subintervals into smaller pieces without removing any.

The key idea is to estimate the contribution of $f$ on each subinterval by its extreme values. Since $f$ is bounded, the supremum and infimum on each subinterval are finite, and the resulting sums bracket the true area.

[definition:Upper And Lower Darboux Sums] Let $f: [a, b] \to \mathbb{R}$ be a bounded function and $P = \{x_0, \ldots, x_n\}$ a partition of $[a, b]$. For each $i \in \{1, \ldots, n\}$, define \begin{align*} M_i &:= \sup_{x \in [x_{i-1}, x_i]} f(x), \\ m_i &:= \inf_{x \in [x_{i-1}, x_i]} f(x). \end{align*} The **upper Darboux sum** and **lower Darboux sum** of $f$ with respect to $P$ are \begin{align*} U(f, P) &:= \sum_{i=1}^n M_i \, \Delta x_i, \\ L(f, P) &:= \sum_{i=1}^n m_i \, \Delta x_i. \end{align*} [/definition]

For any partition $P$, the inequality $m_i \le M_i$ on each subinterval gives $L(f, P) \le U(f, P)$. More importantly, refinement improves the approximation: if $Q$ is a refinement of $P$, then $L(f, P) \le L(f, Q)$ and $U(f, Q) \le U(f, P)$, because splitting a subinterval can only raise the infimum and lower the supremum. This monotonicity is what makes the following definition possible.

[definition:Upper And Lower Darboux Integrals] Let $f: [a, b] \to \mathbb{R}$ be bounded. The **upper Darboux integral** and **lower Darboux integral** of $f$ are \begin{align*} \overline{\int_a^b} f(x)\, dx &:= \inf_P \, U(f, P), \\ \underline{\int_a^b} f(x)\, dx &:= \sup_P \, L(f, P), \end{align*} where the infimum and supremum are taken over all partitions $P$ of $[a, b]$. [/definition]

The upper integral squeezes down from above by taking the tightest possible overshoot, and the lower integral pushes up from below by taking the largest possible undershoot. For any two partitions $P$ and $Q$, their common refinement $P \cup Q$ satisfies $L(f, P) \le L(f, P \cup Q) \le U(f, P \cup Q) \le U(f, Q)$, so every lower sum is at most every upper sum. Taking the supremum over $P$ on the left and the infimum over $Q$ on the right yields the fundamental inequality

\begin{align*} \underline{\int_a^b} f(x)\, dx \le \overline{\int_a^b} f(x)\, dx. \end{align*}

The two integrals need not be equal — but when they are, the function is integrable.

[definition:Riemann Integrability] A bounded function $f: [a, b] \to \mathbb{R}$ is **Riemann integrable** (or **Darboux integrable**) if \begin{align*} \underline{\int_a^b} f(x)\, dx = \overline{\int_a^b} f(x)\, dx. \end{align*} The common value is the **Riemann integral** of $f$ over $[a, b]$, denoted \begin{align*} \int_a^b f(x)\, dx. \end{align*} The set of all Riemann integrable functions on $[a, b]$ is denoted $\mathcal{R}([a, b])$. [/definition]

[remark:Riemann Sums Versus Darboux Sums] There is a second, historically prior, formulation: a **Riemann sum** chooses sample points $t_i \in [x_{i-1}, x_i]$ and forms $S(f, P, t) = \sum_{i=1}^n f(t_i) \, \Delta x_i$. One declares $f$ integrable if these sums converge to a common limit as $\|P\| \to 0$, regardless of the choice of sample points. The Darboux formulation (via upper and lower sums) and the Riemann formulation are equivalent — the Darboux approach is technically cleaner because it avoids quantifying over sample points, but the Riemann sum formulation is indispensable for numerical computation and for recognising limits as integrals. [/remark]

## The Darboux Criterion

The definition of Riemann integrability asks for the upper and lower integrals to coincide, but checking this directly requires computing two global extrema over all partitions. A much more useful characterisation reduces integrability to a single quantitative condition: for every $\varepsilon > 0$, there exists a partition making the gap between the upper and lower sums smaller than $\varepsilon$. This reformulation is the workhorse behind every integrability proof.

[quotetheorem:281]

The forward direction is immediate from the definitions: if $\overline{\int} f = \underline{\int} f$, then for any $\varepsilon > 0$ one can find partitions $P_1, P_2$ with $U(f, P_1) < \overline{\int} f + \varepsilon/2$ and $L(f, P_2) > \underline{\int} f - \varepsilon/2$, and their common refinement $P = P_1 \cup P_2$ satisfies $U(f, P) - L(f, P) < \varepsilon$. The reverse direction is equally direct: if $U(f, P) - L(f, P) < \varepsilon$ for some $P$, then $\overline{\int} f - \underline{\int} f \le U(f, P) - L(f, P) < \varepsilon$, and since $\varepsilon$ is arbitrary, the two integrals coincide.