For bounded $f$ on $[a,b]$ and a partition $P$: the infimum $m_i$ and supremum $M_i$ of $f$ on the $i$-th subinterval, and the lower and upper Darboux sums $L(f,P) = \sum_i m_i \Delta_i$ and $U(f,P) = \sum_i M_i \Delta_i$

Let $a < b$ be reals, let $f : [a,b] \to \mathbb{R}$ be bounded (Lower bound, bounded below, bounded set), so that there is a real $M \ge 0$ with $|f(x)| \le M$ for every $x \in [a,b]$, and let $P = (n,t)$ be a partition of $[a,b]$ with subintervals $I_i = [t_i, t_{i+1}]$ and lengths $\Delta_i = t_{i+1} - t_i$ for $i < n$ (Partition of $[a,b]$ as a finite strictly increasing list $a = t_0 < t_1 < \dots < t_n = b$, its subintervals and their lengths, its mesh, refinement, and the common refinement of two partitions).

The two extreme values on a subinterval

For $i < n$ put

$m_i \;:=\; \inf f[I_i], \qquad M_i \;:=\; \sup f[I_i], \qquad f[I_i] \;=\; \{\, f(x) : x \in I_i \,\} .$

Both exist. The set $f[I_i]$ is nonempty, because $t_i < t_{i+1}$ makes $I_i$ nonempty (Intervals of $\mathbb{R}$: the nine order-convex forms, nondegeneracy, and length), and it is bounded, because $|f(x)| \le M$ for every $x$ (Lower bound, bounded below, bounded set). A nonempty set bounded above has a supremum (Complete ordered field (least-upper-bound property)) and a nonempty set bounded below has an infimum (Every nonempty set bounded below has an infimum, Greatest lower bound (infimum)); each is unique, so the notations $\inf f[I_i]$ and $\sup f[I_i]$ name single real numbers (Suprema and infima are unique).

They bracket the values, and each other. For $x \in I_i$,

$-M \;\le\; m_i \;\le\; f(x) \;\le\; M_i \;\le\; M ,$

the outer inequalities because $-M$ is a lower bound and $M$ an upper bound of $f[I_i]$, and the middle ones by the definitions of infimum and supremum. In particular $m_i \le M_i$ and $M_i - m_i \le 2M$.

The dependence of $m_i$ and $M_i$ on $f$ and on $P$ is suppressed in the notation, as is customary; where two partitions are in play the sums below carry the partition and the extreme values are written out.

The two Darboux sums

$L(f,P) \;:=\; \sum_{i<n} m_i \,\Delta_i, \qquad U(f,P) \;:=\; \sum_{i<n} M_i \,\Delta_i ,$

the finite sums of Finite sums and finite products, by recursion, indexed by $i \in \mathbb{N}$ with $i < n$. Both are real numbers, being finite sums of reals, and

$L(f,P) \;\le\; U(f,P) ,$

by monotonicity of finite sums, clause 4 of Laws of finite sums and finite products, since $m_i \Delta_i \le M_i \Delta_i$ for every $i < n$: multiplying $m_i \le M_i$ by $\Delta_i > 0$ preserves the inequality (Ordered field).

The gap on a subinterval is the oscillation there

For every $i < n$,

$M_i - m_i \;=\; \omega_f(I_i) \;=\; \sup\{\, |f(x) - f(y)| \ : \ x, y \in I_i \,\} ,$

the oscillation of $f$ on the set $I_i$ (The oscillation $\omega_f(S) = \sup\{\,|f(x) - f(y)| : x, y \in S\,\}$ of $f$ on a set and the oscillation $\omega_f(c) = \inf_{\delta > 0} \omega_f(A \cap N_\delta(c))$ at a point, both taken in the extended reals). The supremum is a real number here rather than an extended one, because $f$ is bounded (The oscillation $\omega_f(S) = \sup\{\,|f(x) - f(y)| : x, y \in S\,\}$ of $f$ on a set and the oscillation $\omega_f(c) = \inf_{\delta > 0} \omega_f(A \cap N_\delta(c))$ at a point, both taken in the extended reals, The extended real line $\overline{\mathbb{R}} = \mathbb{R} \cup \{-\infty, +\infty\}$, its order, and the arithmetic that is left undefined). The identity is proved in two inequalities.

The oscillation is at most the gap. For $x, y \in I_i$ both $f(x)$ and $f(y)$ lie in $[m_i, M_i]$, so $f(x) - f(y) \le M_i - m_i$ and $f(y) - f(x) \le M_i - m_i$, whence $|f(x) - f(y)| \le M_i - m_i$ (Basic properties of the absolute value). So $M_i - m_i$ is an upper bound of the set whose supremum is $\omega_f(I_i)$.

The gap is at most the oscillation. Let $\varepsilon > 0$ be real. By the $\varepsilon$-characterisations of the supremum and the infimum (Epsilon characterisation of the supremum, Epsilon characterisation of the infimum) there are $x, y \in I_i$ with $f(x) > M_i - \varepsilon/2$ and $f(y) < m_i + \varepsilon/2$; then

$|f(x) - f(y)| \;\ge\; f(x) - f(y) \;>\; (M_i - m_i) - \varepsilon ,$

so $\omega_f(I_i) > (M_i - m_i) - \varepsilon$. As $\varepsilon > 0$ was arbitrary, $\omega_f(I_i) \ge M_i - m_i$: otherwise $\varepsilon := (M_i - m_i) - \omega_f(I_i)$ would be positive and give $\omega_f(I_i) > \omega_f(I_i)$.

This identity is what connects the Darboux machinery to the pointwise oscillation of The oscillation $\omega_f(S) = \sup\{\,|f(x) - f(y)| : x, y \in S\,\}$ of $f$ on a set and the oscillation $\omega_f(c) = \inf_{\delta > 0} \omega_f(A \cap N_\delta(c))$ at a point, both taken in the extended reals, and it is the hinge of Lebesgue's criterion for Riemann integrability: a bounded $f$ on $[a,b]$ is Riemann integrable if and only if its set of discontinuities has measure zero.

• Boundedness is a hypothesis of the definition, not of the theorems. Without it $f[I_i]$ may fail to have a supremum in $\mathbb{R}$ and $U(f,P)$ is not defined at all. Every statement on this page that mentions $L$ or $U$ therefore carries "bounded $f$" in its hypotheses, and none of them is a restriction that could be lifted: an unbounded function has no Darboux sums to compare.

• Why the infimum and the supremum, and not a value of $f$. Replacing $m_i$ by $f(\xi_i)$ for a point $\xi_i \in I_i$ gives the Riemann sums of Tagged partitions of $[a,b]$, with a tag $\xi_i$ in each subinterval, and the Riemann sum $S(f,P,\xi) = \sum_i f(\xi_i)\,\Delta_i$, which depend on a choice of points and are not extremal. The Darboux sums are canonical functions of $f$ and $P$ alone, which is what makes the supremum and infimum over all partitions in The lower and upper Darboux integrals of a bounded $f$ on $[a,b]$ as $\sup_P L(f,P)$ and $\inf_P U(f,P)$, Darboux integrability as their equality, and the notation $\int_a^b f$ well posed without any selection.

• The empty sum does not occur. A partition has $n \ge 1$ (Partition of $[a,b]$ as a finite strictly increasing list $a = t_0 < t_1 < \dots < t_n = b$, its subintervals and their lengths, its mesh, refinement, and the common refinement of two partitions), so $L(f,P)$ and $U(f,P)$ are sums of at least one term. The first term is $m_0 \Delta_0$, over the subinterval $[a, t_1]$; the indexing starts at $i = 0$ throughout this page.