The Riemann Integral: Definition and Integrability: Examples and Counterexamples

Given: $f : [0,1] \to \mathbb{R}$ with $f(x) = x^{2}$, and for $N \ge 1$ the uniform partition $U_N = (N,t)$ of $[0,1]$ with $t_i = \iota(i)/\iota(N)$ for $i \le N$ and lengths $\Delta_i = 1/\iota(N)$.

For $U_N$: $t_0 = 0$, $t_N = 1$, $t_i < t_{i+1}$, $\Delta_i = 1/\iota(N)$ for $i < N$, $\sum_{i<N}\Delta_i = 1$, and $I_i = [t_i,t_{i+1}]$ (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, Intervals of $\mathbb{R}$: the nine order-convex forms, nondegeneracy, and length).

$m_i = \inf f[I_i]$, $M_i = \sup f[I_i]$, $L(f,P) = \sum_{i<N}m_i\Delta_i$, $U(f,P) = \sum_{i<N}M_i\Delta_i$, and $L(f,P) \le \underline{\int_0^1} f \le \overline{\int_0^1} f \le U(f,P)$ for every partition $P$; when $f$ is integrable, $\int_0^1 f$ is the common value of the two integrals (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 $U(f,P) = \sum_i M_i \Delta_i$, 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 $\int_a^b f$).

For $0 \le a < b$ one has $a^{2} < b^{2}$; a set with a least element has it as its infimum and one with a greatest element has it as its supremum (Monotonicity of $x \mapsto x^n$ and of $n \mapsto a^n$, Integer powers $a^m$, Greatest lower bound (infimum), Maximum and minimum of a set, Complete ordered field (least-upper-bound property)).

The closed form. For every $n \in \mathbb{N}$, $6\sum_{k<n}\iota(k)^{2} = \iota(n)(\iota(n)-1)(2\iota(n)-1)$. This is an induction on $n$ (The principle of mathematical induction) from the recursion clause $\sum_{k<n+1} = \sum_{k<n} + \iota(n)^2$ of Finite sums and finite products, by recursion and the identity $x(x-1)(2x-1) + 6x^{2} = x(x+1)(2x+1)$, valid in any commutative ring and in particular for $x = \iota(n)$ (Ordered field, Canonical naturals are positive and strictly increasing).

Finite sums: scaling, additivity, telescoping $\sum_{i<N}(c_{i+1}-c_i) = c_N - c_0$, and $\iota(i+1) = \iota(i)+1$ (Finite sums and finite products, by recursion, Laws of finite sums and finite products, The canonical natural $\iota(n) = n \cdot 1_F$ of a field, Canonical naturals are positive and strictly increasing).

$x \mapsto x^{2}$ is continuous on $[0,1]$, being a polynomial function, and a continuous function on a closed bounded interval with distinct endpoints is Riemann integrable (Sums, scalar multiples, products, absolute values, maxima, minima and quotients with nonvanishing denominator of continuous functions are continuous, as are constants, the identity and every polynomial function, Continuity of $f : A \to \mathbb{R}$ at a point of $A$ and on $A$: the $\varepsilon$-$\delta$ condition, its agreement with $\lim_{x \to c} f(x) = f(c)$ at a limit point, and continuity at an isolated point, A continuous function on $[a,b]$ is Riemann integrable, by Heine-Cantor and Riemann's criterion).

For every real $\eta > 0$ there is a natural $N \ge 1$ with $1/\iota(N) < \eta$, and $\iota(N) > 0$ for $N \ge 1$ (For every $\varepsilon > 0$ in a complete ordered field there is a natural $n \ge 1$ with $1/n < \varepsilon$, Every complete ordered field is Archimedean, The canonical natural $\iota(n) = n \cdot 1_F$ of a field, Canonical naturals are positive and strictly increasing).

Ordered-field arithmetic and the absolute value: adding a constant and multiplying by a positive quantity preserve an inequality; the order is total and transitive; $|u - v| \le w$ whenever $c \le u \le d$, $c \le v \le d$ and $d - c \le w$ (Basic properties of the absolute value, Order is preserved by adding a constant and by adding inequalities, Sign rules for products and monotonicity of multiplication, Ordered field, Complete ordered field (least-upper-bound property)). These order-arithmetic facts are stated by their sources for the strict order only; the nonstrict forms used below follow by adjoining the equality case, in which the two sides coincide.

Given: $f : [0,1] \to \mathbb{R}$ with $f(x) = x^{2}$; the partition $P_0 = (1,t)$ with $t_0 = 0$ and $t_k = 1$ for $k \ge 1$; and $P_1 = (2,s)$ with $s_0 = 0$, $s_1 = 1/2$ and $s_k = 1$ for $k \ge 2$.

Both $P_0$ and $P_1$ are partitions of $[0,1]$, and $P_1$ refines $P_0$, since $\operatorname{pts}(P_0) = \{0,1\} \subseteq \{0,\tfrac12,1\} = \operatorname{pts}(P_1)$; the subintervals of $P_0$ are $[0,1]$ with length $1$, and those of $P_1$ are $[0,\tfrac12]$ and $[\tfrac12,1]$, each of length $\tfrac12$ (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, Intervals of $\mathbb{R}$: the nine order-convex forms, nondegeneracy, and length).

$m_i = \inf f[I_i]$, $M_i = \sup f[I_i]$, $L(f,P) = \sum_{i<n}m_i\Delta_i$ and $U(f,P) = \sum_{i<n}M_i\Delta_i$ (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 $U(f,P) = \sum_i M_i \Delta_i$).

For $0 \le a < b$ one has $a^{2} < b^{2}$, so on an interval $[c,d] \subseteq [0,1]$ with $c < d$ the function $f$ has least value $c^{2}$ and greatest value $d^{2}$, both attained; a set with a least element has it as its infimum and one with a greatest element has it as its supremum (Monotonicity of $x \mapsto x^n$ and of $n \mapsto a^n$, Integer powers $a^m$, Greatest lower bound (infimum), Maximum and minimum of a set, Complete ordered field (least-upper-bound property)).

If $P'$ refines $P$ then $L(f,P) \le L(f,P') \le U(f,P') \le U(f,P)$ (Refining a partition raises the lower Darboux sum and lowers the upper one, and every lower sum is at most every upper sum: $L(f,P) \le L(f,P') \le U(f,P') \le U(f,P)$ when $P'$ refines $P$, and $L(f,P) \le U(f,Q)$ for arbitrary partitions $P$ and $Q$; moreover the two changes are at most $2M(n' - n)\|P\|$).

Finite sums of one and of two terms: $\sum_{i<1}a_i = a_0$ and $\sum_{i<2}a_i = a_0 + a_1$ (Finite sums and finite products, by recursion, Laws of finite sums and finite products).

Ordered-field arithmetic: $(\tfrac12)^{2} = \tfrac14$, $\tfrac14 \cdot \tfrac12 = \tfrac18$, $\tfrac18 + \tfrac12 = \tfrac58$, and $0 < \tfrac18 < \tfrac58 < 1$ (Order is preserved by adding a constant and by adding inequalities, Sign rules for products and monotonicity of multiplication, Ordered field, Complete ordered field (least-upper-bound property)). These order-arithmetic facts are stated by their sources for the strict order only; the nonstrict forms used below follow by adjoining the equality case, in which the two sides coincide.

Given: $g : [0,3] \to \mathbb{R}$ with $g(x) = \lfloor x \rfloor$; a natural $M \ge 1$; $N := 3M$; and the uniform partition $U_N = (N,t)$ of $[0,3]$ with $t_i = 3\iota(i)/\iota(N) = \iota(i)/\iota(M)$ for $i \le N$ and lengths $\Delta_i = 3/\iota(N) = 1/\iota(M)$.

For every real $x$ there is exactly one integer $\lfloor x \rfloor$ with $\lfloor x \rfloor \le x < \lfloor x \rfloor + 1$ (Integer part: for every real $x$ there is exactly one integer $m$ with $m \le x < m + 1$).

$g$ is nondecreasing: for $x \le y$, $\lfloor x \rfloor \le x \le y < \lfloor y \rfloor + 1$, and $\lfloor x \rfloor$, $\lfloor y \rfloor$ are integers, so $\lfloor x \rfloor \le \lfloor y \rfloor$, no integer lying strictly between $\lfloor y \rfloor$ and $\lfloor y \rfloor + 1$ (Integer part: for every real $x$ there is exactly one integer $m$ with $m \le x < m + 1$, Nondecreasing, increasing (strictly increasing), nonincreasing, decreasing, monotone and strictly monotone real functions on a subset of $\mathbb{R}$, with the dictionary to monotone sequences).

A monotone function on a closed bounded interval with distinct endpoints is bounded and Riemann integrable (A monotone function on $[a,b]$ is Riemann integrable: for the uniform partition into $N$ parts the upper minus lower sum telescopes to $|f(b) - f(a)|\,(b-a)/\iota(N)$, 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 $\int_a^b f$).

For $U_N$: $t_0 = 0$, $t_N = 3$, $t_i < t_{i+1}$, $\Delta_i = 1/\iota(M)$, $\sum_{i<N}\Delta_i = 3$ (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, Intervals of $\mathbb{R}$: the nine order-convex forms, nondegeneracy, and length).

For a nondecreasing $g$ and a subinterval $I_i = [t_i,t_{i+1}]$: $m_i = g(t_i)$ and $M_i = g(t_{i+1})$, both attained; $L(g,P) = \sum_{i<N}m_i\Delta_i$, $U(g,P) = \sum_{i<N}M_i\Delta_i$, and $L(g,P) \le \underline{\int_0^3}g \le \overline{\int_0^3}g \le U(g,P)$ (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 $U(f,P) = \sum_i M_i \Delta_i$, 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 $\int_a^b f$, Greatest lower bound (infimum), Maximum and minimum of a set, Complete ordered field (least-upper-bound property)).

Finite sums: splitting a sum over $i < 3M$ into the three blocks $i < M$, $M \le i < 2M$ and $2M \le i < 3M$; scaling; and $\sum_{i<M}c = \iota(M)c$ (Finite sums and finite products, by recursion, Laws of finite sums and finite products).

For $i \in \mathbb{N}$: $\lfloor \iota(i)/\iota(M) \rfloor = 0$ when $i < M$, $= 1$ when $M \le i < 2M$, and $= 2$ when $2M \le i < 3M$; and $\lfloor \iota(j)/\iota(M)\rfloor = 3$ for $j = 3M$. Each case is [L1] applied to the displayed inequalities $0 \le \iota(i)/\iota(M) < 1$, $1 \le \iota(i)/\iota(M) < 2$, $2 \le \iota(i)/\iota(M) < 3$ and $\iota(3M)/\iota(M) = 3$, which follow from $\iota$ being strictly increasing and additive (Canonical naturals are positive and strictly increasing, The canonical natural $\iota(n) = n \cdot 1_F$ of a field, Order is preserved by adding a constant and by adding inequalities). These order-arithmetic facts are stated by their sources for the strict order only; the nonstrict forms used here follow by adjoining the equality case, in which the two sides coincide.

For every real $\eta > 0$ there is a natural $M \ge 1$ with $1/\iota(M) < \eta$, and $\iota(M) > 0$ for $M \ge 1$ (For every $\varepsilon > 0$ in a complete ordered field there is a natural $n \ge 1$ with $1/n < \varepsilon$, Every complete ordered field is Archimedean, The canonical natural $\iota(n) = n \cdot 1_F$ of a field, Canonical naturals are positive and strictly increasing).

Ordered-field arithmetic and the absolute value: adding a constant and multiplying by a positive quantity preserve an inequality; the order is total and transitive (Basic properties of the absolute value, Order is preserved by adding a constant and by adding inequalities, Sign rules for products and monotonicity of multiplication, Ordered field, Complete ordered field (least-upper-bound property)). These order-arithmetic facts are stated by their sources for the strict order only; the nonstrict forms used below follow by adjoining the equality case, in which the two sides coincide.