FALSE: every bounded function on $[a,b]$ is Riemann integrable

Given: The Dirichlet function $\mathbf{1}_{\mathbb{Q}}$ restricted to $[0,1]$, that is $g : [0,1] \to \mathbb{R}$ with $g(x) = 1$ for rational $x$ and $g(x) = 0$ for irrational $x$ (The Dirichlet function $1_{\mathbb{Q}}$, and Thomae's function $t$ with $t(x) = 1/q$ at a rational $x = p/q$ in lowest terms with $q \ge 1$ and $t(x) = 0$ at every irrational $x$).

The false claim: every bounded function on a closed bounded interval with distinct endpoints is Riemann integrable.

Both $\mathbb{Q}$ and $\mathbb{R}\setminus\mathbb{Q}$ are dense in $\mathbb{R}$, and a set is dense exactly when every neighbourhood of every real meets it (Both $\mathbb{Q}$ and $\mathbb{R} \setminus \mathbb{Q}$ are dense in $\mathbb{R}$, and every nonempty open subset of $\mathbb{R}$ is uncountable, The closure equals the set together with its limit points, equals the set of points every neighbourhood of which meets it, and is the smallest closed superset; a set is closed iff it contains its limit points, Interior, closure, boundary and exterior of a subset of $\mathbb{R}$, The $\varepsilon$-neighbourhood and the punctured $\varepsilon$-neighbourhood of a point of $\mathbb{R}$).

For a partition $P = (n,t)$ of $[0,1]$: $n \ge 1$, $t_i < t_{i+1}$, $\Delta_i = t_{i+1}-t_i > 0$, $\sum_{i<n}\Delta_i = 1 - 0 = 1$, 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).

$m_i = \inf g[I_i]$, $M_i = \sup g[I_i]$, $L(g,P) = \sum_{i<n}m_i\Delta_i$, $U(g,P) = \sum_{i<n}M_i\Delta_i$; $\underline{\int_0^1} g$ is the supremum of the lower sums and $\overline{\int_0^1} g$ the infimum of the upper sums; $g$ is integrable exactly when these agree (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 $\int_a^b f$).

A set with a least element has it as its infimum, and one with a greatest element has it as its supremum; the supremum and infimum of $\{c\}$ are both $c$ (Greatest lower bound (infimum), Maximum and minimum of a set, Complete ordered field (least-upper-bound property)).

Finite sums: scaling and $\sum_{i<n}0 = 0$ (Finite sums and finite products, by recursion, Laws of finite sums and finite products).

Ordered-field arithmetic: the midpoint $(u+v)\cdot 2^{-1}$ of $u < v$ satisfies $u < (u+v)\cdot 2^{-1} < v$, and $N_\rho(c) \subseteq (u,v)$ for $c$ the midpoint and $\rho := (v-u)\cdot 2^{-1}$ (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), The $\varepsilon$-neighbourhood and the punctured $\varepsilon$-neighbourhood of a point of $\mathbb{R}$). 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.