A curve drawn by a physical instrument rarely comes with a formula for an antiderivative. The older problem of integration is more basic: how should we assign an area to a [bounded function](/page/Bounded%20Function) on an interval using only finite measurements? If we sample the function at finitely many points, we may miss spikes. If we use rectangles, the answer depends on where the rectangles touch the graph. Riemann integrability is the theory that explains when these finite rectangle approximations settle down to a single number.

The central tension is that a function can be bounded and still behave badly at many points. The [Riemann integral](/page/Riemann%20Integral) tolerates discontinuities when their total effect can be confined to intervals of small total length, but it rejects functions whose oscillation remains visible across a set too large to ignore. This makes Riemann integrability a special case of integrability: it is not the most general integral, but it is the one built from partitions of intervals and upper and lower rectangular estimates.

[example: A Function That Defeats Sampling] Let $f: [0,1] \to \mathbb{R}$ be defined by $f(x)=1$ for $x \in \mathbb{Q}$ and $f(x)=0$ for $x \notin \mathbb{Q}$. We show that no partition can make the upper and lower rectangular estimates approach one another. Let $P=(x_0,x_1,\ldots,x_n)$ be any partition of $[0,1]$. For each $i \in \{1,\ldots,n\}$, the interval $[x_{i-1},x_i]$ has positive length, so it contains at least one rational point and at least one irrational point. Hence the set of values of $f$ on $[x_{i-1},x_i]$ contains both $0$ and $1$, and since $0 \le f(x) \le 1$ for every $x \in [0,1]$, we have \begin{align*} m_i(f,P)=\inf\{f(x):x \in [x_{i-1},x_i]\}=0. \end{align*} Similarly, \begin{align*} M_i(f,P)=\sup\{f(x):x \in [x_{i-1},x_i]\}=1. \end{align*} Therefore the lower sum is \begin{align*} L(P,f)=\sum_{i=1}^n 0\cdot (x_i-x_{i-1})=0. \end{align*} The upper sum is \begin{align*} U(P,f)=\sum_{i=1}^n 1\cdot (x_i-x_{i-1})=\sum_{i=1}^n (x_i-x_{i-1}). \end{align*} The subinterval lengths telescope: \begin{align*} \sum_{i=1}^n (x_i-x_{i-1})=(x_1-x_0)+(x_2-x_1)+\cdots+(x_n-x_{n-1})=x_n-x_0=1. \end{align*} Thus every partition satisfies \begin{align*} U(P,f)-L(P,f)=1-0=1. \end{align*} Since this gap never becomes smaller than $1$, it cannot be made less than any $\varepsilon$ with $0<\varepsilon<1$. The obstruction is not unboundedness, because $0 \le f \le 1$; it is that the oscillation between rational and irrational points persists on every subinterval. [/example]

This example is the right warning at the start. The issue is not the size of the function: $f$ is bounded by $1$. The issue is persistent oscillation. Riemann integration asks whether refining a partition can make the total uncertainty between upper and lower rectangles as small as desired.

## Definition

The page is about one central question: when do finite rectangle estimates determine a unique area? We state the primary concept first, using the Darboux gap as the defining test. The supporting objects that make the notation precise are unpacked immediately afterward.

[definition: Riemann Integrable Function] Let $a,b \in \mathbb{R}$ with $a < b$, and let $f: [a,b] \to \mathbb{R}$ be bounded. The function $f$ is Riemann integrable on $[a,b]$ if for every $\varepsilon > 0$ there exists a partition $P$ of $[a,b]$ such that \begin{align*} U(P,f)-L(P,f) < \varepsilon, \end{align*} where $U(P,f)$ and $L(P,f)$ are the upper and lower sums of $f$ with respect to $P$. [/definition]

This definition is deliberately narrow: bounded function, compact interval, finite partitions. Its strength is that it turns a geometric idea into a precise finite-approximation test. Its limitation is that it does not cover many functions that modern measure theory can integrate.

### Partitions and Mesh

The definition refers to partitions, so we now make the finite observation scheme precise. A partition records the finitely many vertical cuts where the interval is inspected. The more refined the partition, the more local information the upper and lower rectangles can use.

[definition: Partition of a Compact Interval] Let $a,b \in \mathbb{R}$ with $a < b$. A partition of $[a,b]$ is a finite ordered tuple \begin{align*} P = (x_0,x_1,\ldots,x_n) \end{align*} such that \begin{align*} a = x_0 < x_1 < \cdots < x_n = b. \end{align*} The subintervals of $P$ are $[x_{i-1},x_i]$ for $i \in \{1,\ldots,n\}$. [/definition]

A partition by itself only names intervals. To compare coarse and fine observations, we need a number measuring the largest unresolved interval left by the partition; that number controls the worst local oscillation that a rectangle may hide.

[definition: Mesh of a Partition] Let $a,b \in \mathbb{R}$ with $a < b$, and let $\mathcal{P}([a,b])$ be the set of partitions of $[a,b]$. The mesh is the map \begin{align*} |\cdot|: \mathcal{P}([a,b]) &\to \mathbb{R}. \end{align*} For $P=(x_0,x_1,\ldots,x_n) \in \mathcal{P}([a,b])$, define \begin{align*} |P| = \max_{1 \le i \le n}(x_i-x_{i-1}). \end{align*} [/definition]

The mesh is useful when comparing Riemann sums along sequences of partitions. The Darboux definition below uses all partitions at once, but the mesh formulation later explains why ordinary rectangle approximations converge to the same value. To define those Darboux estimates, we now need the best lower and upper rectangle heights on each interval.

### Upper and Lower Estimates

A bounded function has a highest and lowest visible value on each subinterval after replacing maximum and minimum by [supremum and infimum](/page/Supremum%20and%20Infimum). These are the rectangle heights that give the best possible upper and lower estimates relative to a chosen partition.

[definition: Upper and Lower Sums] Let $a,b \in \mathbb{R}$ with $a < b$. Let $\mathcal{B}([a,b])$ be the set of bounded functions $f: [a,b] \to \mathbb{R}$, and let $\mathcal{P}([a,b])$ be the set of partitions of $[a,b]$. The upper-sum functional and lower-sum functional are maps \begin{align*} U,L: \mathcal{P}([a,b]) \times \mathcal{B}([a,b]) &\to \mathbb{R}. \end{align*} For $f \in \mathcal{B}([a,b])$ and $P=(x_0,x_1,\ldots,x_n) \in \mathcal{P}([a,b])$, define $U(P,f)$ and $L(P,f)$ as follows. For each $i \in \{1,\ldots,n\}$, set \begin{align*} M_i(f,P) = \sup\{f(x):x \in [x_{i-1},x_i]\}. \end{align*} Set \begin{align*} m_i(f,P) = \inf\{f(x):x \in [x_{i-1},x_i]\}. \end{align*} The upper sum of $f$ with respect to $P$ is \begin{align*} U(P,f) = \sum_{i=1}^n M_i(f,P)(x_i-x_{i-1}). \end{align*} The lower sum of $f$ with respect to $P$ is \begin{align*} L(P,f) = \sum_{i=1}^n m_i(f,P)(x_i-x_{i-1}). \end{align*} [/definition]

The upper sum is the cost of covering the graph from above using rectangles adapted to $P$; the lower sum is the guaranteed area captured from below. The integral should exist exactly when the gap between what is forced and what is possible can be squeezed away, so we take the best upper and lower estimates over all finite partitions.

[definition: Upper and Lower Riemann Integrals] Let $a,b \in \mathbb{R}$ with $a < b$, and let $\mathcal{B}([a,b])$ be the set of bounded functions $f: [a,b] \to \mathbb{R}$. The upper Riemann integral and lower Riemann integral are functionals \begin{align*} \overline{I}_{[a,b]},\underline{I}_{[a,b]}: \mathcal{B}([a,b]) &\to \mathbb{R}. \end{align*} For $f \in \mathcal{B}([a,b])$, the upper Riemann integral of $f$ over $[a,b]$ is \begin{align*} \overline{I}_{[a,b]}(f) = \inf\{U(P,f):P \text{ is a partition of } [a,b]\}. \end{align*} The lower Riemann integral of $f$ over $[a,b]$ is \begin{align*} \underline{I}_{[a,b]}(f) = \sup\{L(P,f):P \text{ is a partition of } [a,b]\}. \end{align*} [/definition]

There is always an inequality $\underline{I}_{[a,b]}(f) \le \overline{I}_{[a,b]}(f)$, because every lower sum lies below every upper sum after passing to a common refinement. The theory becomes interesting when this inequality collapses to equality: then all sufficiently fine rectangular estimates are being forced toward one number rather than toward an interval of possible values.