--- title: Partitions, Darboux Sums, and Integrability module: The Riemann Integral moduleNumber: 6 lessonNumber: 1 order: 601 summary: > The Riemann integral is defined by trapping the area under a bounded function between under- and over-estimates. Partitions cut the domain into strips; lower and upper Darboux sums bracket the area; refining a partition tightens the bracket. A function is integrable exactly when the bracket can be made arbitrarily thin, and the tagged Riemann-sum limit gives the same number. topics: [The Riemann Integral] draft: false sources: - book: Lebl ref: "Ch. 5 — The Riemann Integral; §5.1 The Riemann integral" - book: Shkoller ref: "§1.1 Partitions and Riemann sums; §1.3 Upper and Lower Riemann Sums; §1.4 Refinement; §1.5 Properties of upper and lower sums; §1.6 Definition equivalence" --- 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](/real-analysis/riemann-integration/fundamental-theorem). 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.[^lebl-51] Throughout, $f : [a,b] \to \mathbb{R}$ is **bounded** — unbounded functions are handled separately as [improper integrals](/real-analysis/riemann-integration/log-exp-improper). ## 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. > **Definition (Partition).** A **partition** $P$ of $[a,b]$ is a finite set > $\{x_0, x_1, \dots, x_n\}$ with > $$ > a = x_0 < x_1 < x_2 < \cdots < x_{n-1} < x_n = b. > $$ > Write $\Delta x_i \coloneqq x_i - x_{i-1}$ for the width of the $i$-th > subinterval $[x_{i-1}, x_i]$. The largest width $\max_i \Delta x_i$ is the > **mesh** of $P$. > **Definition (Darboux sums).** For a partition $P$ of $[a,b]$, set > $$ > m_i \coloneqq \inf\{ f(x) : x \in [x_{i-1}, x_i] \}, > \qquad > M_i \coloneqq \sup\{ f(x) : x \in [x_{i-1}, x_i] \}. > $$ > The **lower Darboux sum** and **upper Darboux sum** are > $$ > L(P, f) \coloneqq \sum_{i=1}^n m_i \, \Delta x_i, > \qquad > U(P, f) \coloneqq \sum_{i=1}^n M_i \, \Delta x_i. > $$ 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. $$ % caption: Lower and upper Darboux sums for an increasing $f$ on a four-piece % caption: partition. Shaded blocks have height $m_i$, the infimum on each % caption: subinterval; the taller outlines rise to $M_i$, the supremum. Their % caption: total-area difference is the unshaded region. \begin{tikzpicture}[scale=1.0, >=stealth, font=\small] \definecolor{acc}{HTML}{4A6FA5} % upper rectangles (outline) \draw[black] (0,0) rectangle (1.2,1.5); \draw[black] (1.2,0) rectangle (2.4,2.3); \draw[black] (2.4,0) rectangle (3.6,2.9); \draw[black] (3.6,0) rectangle (4.8,3.3); % lower rectangles (shaded) \fill[acc!14, draw=acc] (0,0) rectangle (1.2,0.5); \fill[acc!14, draw=acc] (1.2,0) rectangle (2.4,1.5); \fill[acc!14, draw=acc] (2.4,0) rectangle (3.6,2.3); \fill[acc!14, draw=acc] (3.6,0) rectangle (4.8,2.9); % the graph (increasing, piecewise linear through the sample heights) \draw[acc, very thick] (0,0.5) -- (1.2,1.5) -- (2.4,2.3) -- (3.6,2.9) -- (4.8,3.3); \node[acc] at (4.5,3.55) {$y = f(x)$}; % height labels on the third strip \node[acc, font=\footnotesize, anchor=east] at (2.4,2.15) {$m_3$}; \node[black, font=\footnotesize, anchor=east] at (2.4,2.75) {$M_3$}; % axes and ticks \draw[->] (-0.15,0) -- (5.25,0); \draw[->] (0,-0.15) -- (0,3.8); \foreach \x/\lab in {0/x_0, 1.2/x_1, 2.4/x_2, 3.6/x_3, 4.8/x_4} \draw (\x,0.07) -- (\x,-0.07) node[below, font=\footnotesize] {$\lab$}; \end{tikzpicture} $$ 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. > **Proposition (Sums are bounded).** If $m \le f(x) \le M$ for all > $x \in [a,b]$, then for every partition $P$, > $$ > m(b-a) \;\le\; L(P,f) \;\le\; U(P,f) \;\le\; M(b-a). > $$ > **Proof.** A term-by-term comparison: $\sum_i m \, \Delta x_i = m(b-a)$ bounds > $L(P,f)$ below, $\sum_i M \, \Delta x_i = M(b-a)$ bounds $U(P,f)$ above, and > $m_i \le M_i$ gives the middle inequality.[^lebl-51] 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. > **Definition (Darboux integrals).** For bounded $f : [a,b] \to \mathbb{R}$, > the **lower** and **upper Darboux integrals** are > $$ > \underline{\int_a^b} f \coloneqq \sup\{ L(P,f) : P \text{ a partition of } [a,b] \}, > $$ > $$ > \overline{\int_a^b} f \coloneqq \inf\{ U(P,f) : P \text{ a partition of } [a,b] \}. > $$ 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. > **Worked example (Dirichlet function).** Let $f : [0,1] \to \mathbb{R}$ be $f(x) = 1$ > for $x \in \mathbb{Q}$ and $f(x) = 0$ otherwise. Every subinterval contains both > a rational and an irrational, so $m_i = 0$ and $M_i = 1$ for every $i$ and > every partition. Hence $L(P,f) = 0$ and $U(P,f) = 1$ for all $P$, giving > $$ > \underline{\int_0^1} f = 0, \qquad \overline{\int_0^1} f = 1. > $$ 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](/real-analysis/riemann-integration/integrability-classes). ## 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. > **Definition (Refinement).** A partition $\tilde{P}$ is a **refinement** of $P$ > if $P \subseteq \tilde{P}$ as sets — $\tilde{P}$ contains every point of $P$ > and possibly more. 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. $$ % caption: Refining one subinterval by inserting a point. The single inscribed % caption: rectangle (left) is replaced by two taller ones (right); the added % caption: shaded sliver is the increase in the lower sum. Upper sums decrease by % caption: the mirror argument. \begin{tikzpicture}[scale=1.0, >=stealth, font=\small] \definecolor{acc}{HTML}{4A6FA5} % ---- left: coarse ---- \begin{scope} \draw[acc, very thick] (0,0.4) -- (1.5,1.6) -- (3,2.6); \fill[acc!14, draw=acc] (0,0) rectangle (3,0.4); \draw[->] (-0.15,0) -- (3.3,0); \draw[->] (0,-0.15) -- (0,3.0); \foreach \x/\lab in {0/x_{i-1}, 3/x_i} \draw (\x,0.07) -- (\x,-0.07) node[below, font=\footnotesize] {$\lab$}; \node[font=\footnotesize] at (1.5,-0.75) {one rectangle}; \end{scope} % ---- right: refined ---- \begin{scope}[xshift=5cm] \draw[acc, very thick] (0,0.4) -- (1.5,1.6) -- (3,2.6); \fill[acc!14, draw=acc] (0,0) rectangle (1.5,0.4); \fill[acc!22, draw=acc] (1.5,0) rectangle (3,1.6); \draw[->] (-0.15,0) -- (3.3,0); \draw[->] (0,-0.15) -- (0,3.0); \foreach \x/\lab in {0/x_{i-1}, 1.5/t, 3/x_i} \draw (\x,0.07) -- (\x,-0.07) node[below, font=\footnotesize] {$\lab$}; \node[font=\footnotesize] at (1.5,-0.75) {split at $t$}; \end{scope} \end{tikzpicture} $$ > **Proposition (Refinement tightens the bracket).** Let $f$ be bounded and > $\tilde{P}$ a refinement of $P$. Then > $$ > L(P,f) \;\le\; L(\tilde{P}, f) > \qquad\text{and}\qquad > U(\tilde{P}, f) \;\le\; U(P,f). > $$ > **Proof.** By induction it suffices to add a single point $t$ into some > subinterval $[x_{k-1}, x_k]$. Only that term of the sum changes. Writing > $m' = \inf_{[x_{k-1}, t]} f$ and $m'' = \inf_{[t, x_k]} f$, both are at least > $m_k = \inf_{[x_{k-1}, x_k]} f$, so > $$ > m_k \, \Delta x_k = m_k(t - x_{k-1}) + m_k(x_k - t) > \;\le\; m'(t - x_{k-1}) + m''(x_k - t), > $$ > which is the claimed inequality $L(P,f) \le L(\tilde{P},f)$; the upper-sum > inequality is the mirror image.[^shk-15] 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 $$ L(P_1, f) \;\le\; L(\tilde{P}, f) \;\le\; U(\tilde{P}, f) \;\le\; U(P_2, f). $$ Taking the supremum over $P_1$ and the infimum over $P_2$ separates the two Darboux integrals in the right order. > **Proposition (Lower integral below upper).** For bounded > $f : [a,b] \to \mathbb{R}$ and any $m \le f \le M$, > $$ > m(b-a) \;\le\; \underline{\int_a^b} f \;\le\; \overline{\int_a^b} f \;\le\; M(b-a). > $$ ## The Riemann integral When the best under- and over-estimates coincide, that common value is the integral. > **Definition (Riemann integral).** A bounded $f : [a,b] \to \mathbb{R}$ is > **Riemann integrable** if > $$ > \underline{\int_a^b} f = \overline{\int_a^b} f, > $$ > and the common value is the **Riemann integral** $\int_a^b f$. The set of such > functions is $\mathcal{R}[a,b]$. 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. > **Theorem (Cauchy criterion).** A bounded $f : [a,b] \to \mathbb{R}$ is > Riemann integrable if and only if for every $\varepsilon > 0$ there exists a > partition $P$ with > $$ > U(P,f) - L(P,f) < \varepsilon. > $$ > **Proof.** If such a $P$ exists for every $\varepsilon$, then > $0 \le \overline{\int} f - \underline{\int} f \le U(P,f) - L(P,f) < \varepsilon$, > forcing the two integrals equal.[^lebl-51] Conversely, if $f$ is integrable, > pick $P_1$ with $L(P_1,f)$ within $\varepsilon/2$ of the integral and $P_2$ > with $U(P_2,f)$ within $\varepsilon/2$; their common refinement does both at > once.[^shk-19] 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 $\varepsilon$. $$ % caption: The gap $U(P,f) - L(P,f)$ is the total area of the unshaded caps. On a % caption: coarse partition (left) it is large; refining (right) shrinks each cap, % caption: and integrability means the total can be pushed below any $\varepsilon$. \begin{tikzpicture}[scale=1.0, >=stealth, font=\small] \definecolor{acc}{HTML}{4A6FA5} % ---- left: coarse, 3 strips ---- \begin{scope} \draw[acc, very thick] (0,0.5) -- (1.4,1.5) -- (2.8,2.2) -- (4.2,2.7); \foreach \a/\b/\lo/\hi in {0/1.4/0.5/1.5, 1.4/2.8/1.5/2.2, 2.8/4.2/2.2/2.7}{ \fill[acc!14] (\a,0) rectangle (\b,\lo); \fill[black] (\a,\lo) rectangle (\b,\hi); \draw[black] (\a,0) rectangle (\b,\hi); } \draw[->] (-0.15,0) -- (4.6,0); \draw[->] (0,-0.15) -- (0,3.1); \node[font=\footnotesize] at (2.1,-0.55) {coarse}; \end{scope} % ---- right: fine, 6 strips ---- \begin{scope}[xshift=5.6cm] \draw[acc, very thick] (0,0.5) -- (0.7,1.05) -- (1.4,1.5) -- (2.1,1.88) -- (2.8,2.2) -- (3.5,2.47) -- (4.2,2.7); \foreach \a/\lo/\hi in {0/0.5/1.05, 0.7/1.05/1.5, 1.4/1.5/1.88, 2.1/1.88/2.2, 2.8/2.2/2.47, 3.5/2.47/2.7}{ \fill[acc!14] (\a,0) rectangle ({\a+0.7},\lo); \fill[black] (\a,\lo) rectangle ({\a+0.7},\hi); \draw[black] (\a,0) rectangle ({\a+0.7},\hi); } \draw[->] (-0.15,0) -- (4.6,0); \draw[->] (0,-0.15) -- (0,3.1); \node[font=\footnotesize] at (2.1,-0.55) {denser}; \end{scope} \end{tikzpicture} $$ ## Worked examples > **Worked example (A constant is integrable).** If $f \equiv c$, take > $m = M = c$; the bounding proposition forces > $\underline{\int} = \overline{\int} = c(b-a)$, so $\int_a^b c = c(b-a)$. > **Worked example (A single jump).** Let $f : [0,2] \to \mathbb{R}$ equal $1$ > for $x < 1$, $\tfrac12$ at $x = 1$, and $0$ for $x > 1$. On the partition > $P = \{0, 1-\epsilon, 1+\epsilon, 2\}$ the sums are $L(P,f) = 1 - \epsilon$ and > $U(P,f) = 1 + \epsilon$, so $U(P,f) - L(P,f) = 2\epsilon$. The Cauchy criterion > is met, and $\int_0^2 f = 1$.[^lebl-51] > **Worked example (The integral of $x$ and $x^2$).** Take the uniform partition > of $[0,1]$ with $x_i = i/N$, so $\Delta x_i = 1/N$. Since $x \mapsto x$ is > increasing, $m_i = (i-1)/N$ and $M_i = i/N$, and > $$ > L(P_N, f) = \frac{1}{N} \sum_{i=1}^{N} \frac{i-1}{N} = \frac{(N-1)N}{2N^2}, > \qquad > U(P_N, f) = \frac{1}{N} \sum_{i=1}^{N} \frac{i}{N} = \frac{N(N+1)}{2N^2}. > $$ > Both tend to $\tfrac12$ as $N \to \infty$, so $\int_0^1 x \, \d x = \tfrac12$.[^shk-13] > The same partition with $m_i = ((i-1)/N)^2$, $M_i = (i/N)^2$ and the identity > $\sum_{i=1}^N i^2 = \tfrac16 N(N+1)(2N+1)$ gives > $\int_0^1 x^2 \, \d x = \tfrac13$. > **Worked example (A telescoping estimate).** For $f(x) = \frac{1}{1+x}$ on > $[0,b]$, the uniform partition gives a lower/upper gap that telescopes: > $$ > U(P,f) - L(P,f) > = \frac{b}{N}\sum_{i=1}^{N}\left(\frac{1}{1 + x_{i-1}} - \frac{1}{1 + x_i}\right) > = \frac{b}{N}\left(1 - \frac{1}{1+b}\right) > = \frac{b^2}{N(b+1)}. > $$ > Choosing $N$ large makes this less than $\varepsilon$, so $\tfrac{1}{1+x}$ is > integrable on $[0,b]$ without invoking continuity.[^lebl-51] ## 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. > **Definition (Riemann sum).** Given a partition $P$ and a **tag** > $z_i \in [x_{i-1}, x_i]$ in each subinterval, the **Riemann sum** is > $$ > S(P, f) \coloneqq \sum_{i=1}^{n} f(z_i)\,\Delta x_i. > $$ > The function is Riemann integrable with integral $I$ if for every > $\varepsilon > 0$ there is a $\delta > 0$ such that any tagged partition with > mesh below $\delta$ has $|S(P,f) - I| < \varepsilon$, regardless of the tags. 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: $$ L(P,f) \;\le\; S(P,f) \;\le\; U(P,f). $$ $$ % caption: A tagged Riemann sum picks one sample height $f(z_i)$ per strip; its % caption: rectangle sits between the inscribed ($m_i$) and circumscribed ($M_i$) % caption: heights, so the tagged sum lies between $L(P,f)$ and $U(P,f)$. \begin{tikzpicture}[scale=1.0, >=stealth, font=\small] \definecolor{acc}{HTML}{4A6FA5} \draw[black] (1.4,0) rectangle (2.8,2.2); \fill[acc!14, draw=acc] (1.4,0) rectangle (2.8,1.5); \draw[acc, thick, dashed] (1.4,1.85) -- (2.8,1.85); \draw[acc, very thick] (0,0.5) -- (1.4,1.5) -- (2.8,2.2) -- (4.2,2.6); \fill[acc] (2.05,1.85) circle (1.6pt); \node[acc, font=\footnotesize, anchor=south] at (2.05,1.9) {$f(z_i)$}; \draw[->] (-0.15,0) -- (4.5,0); \draw[->] (0,-0.15) -- (0,3.0); \foreach \x/\lab in {1.4/x_{i-1}, 2.05/z_i, 2.8/x_i} \draw (\x,0.07) -- (\x,-0.07) node[below, font=\footnotesize] {$\lab$}; \node[acc, font=\footnotesize, anchor=east] at (1.4,0.75) {$m_i$}; \node[black, font=\footnotesize, anchor=west] at (2.8,2.0) {$M_i$}; \end{tikzpicture} $$ Both definitions describe the same class of functions and the same number. If $U(P,f) - L(P,f) < \varepsilon$, then the tagged sums are trapped in an interval of length $\varepsilon$ 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.[^shk-16] The equivalence is why "Riemann integral" and "Darboux integral" name the same object. | | Darboux definition | Riemann (tagged) definition | | --- | --- | --- | | Rectangle height | $m_i$ (lower), $M_i$ (upper) | $f(z_i)$, arbitrary tag $z_i$ | | Integrability test | $\underline{\int} f = \overline{\int} f$ | mesh limit of $S(P,f)$ exists | | Practical criterion | one $P$ with $U - L < \varepsilon$ | Cauchy: two sums within $\varepsilon$ | | Optimizes over | all partitions (sup / inf) | all partitions and tags (limit) | | Same class? | yes | yes | [^lebl-51]: **Lebl**, _Basic Analysis I_, §5.1 — partitions, lower/upper Darboux sums, the Darboux integrals, Riemann integrability (Def. 5.1.9), and the criterion $U(P,f) - L(P,f) < \varepsilon$ (Prop. 5.1.13), with the constant, step-function, and $\tfrac{1}{1+x}$ examples. [^shk-15]: **Shkoller**, _MAT125B Lecture Notes_, §1.4–1.5 — refinement of partitions and the monotonicity $L_{\delta_1} \le L_{\delta_2} \le U_{\delta_2} \le U_{\delta_1}$ for a refinement. [^shk-19]: **Shkoller**, _MAT125B Lecture Notes_, §1.5 — the Cauchy criterion for integrability in terms of upper and lower sums (Thm. 1.19), and $L(f) \le U(f)$ (Thm. 1.18). [^shk-13]: **Shkoller**, _MAT125B Lecture Notes_, §1.3 — the upper/lower Darboux integrals as $\inf$ and $\sup$ over partitions, with $\int_0^1 x\,\d x = \tfrac12$ and $\int_0^1 x^2\,\d x = \tfrac13$ computed from uniform partitions. [^shk-16]: **Shkoller**, _MAT125B Lecture Notes_, §1.1, §1.6 — the tagged Riemann-sum definition (Def. 1.3–1.4) and the proof that it agrees with the upper/lower-sum definition.