Thomae's function is Riemann integrable on $[0,1]$ with integral $0$: it is continuous at every irrational, so its discontinuity set is countable, and every lower Darboux sum is $0$

Given: Thomae's function $t : [0,1] \to \mathbb{R}$ as above.

$t(x) = 1/\iota(q(x))$ with $\iota(q(x)) \ge 1 > 0$ at a rational $x$, and $t(x) = 0$ at an irrational $x$; hence $0 \le t(x) \le 1$ everywhere and $t(x) > 0$ at every rational $x$ (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 canonical natural $\iota(n) = n \cdot 1_F$ of a field, Canonical naturals are positive and strictly increasing).

Thomae's function on $\mathbb{R}$ is continuous at every irrational and discontinuous at every rational (The Dirichlet function is continuous at no point of $\mathbb{R}$, and Thomae's function is continuous at every irrational and at no rational, so its set of continuity points is exactly the set of irrationals and its oscillation at $c$ equals $t(c)$); a restriction is continuous at every point of the smaller domain at which the original is continuous, the same $\delta$ serving a condition quantified over fewer points (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).

$\mathbb{Q}$ is countably infinite and every subset of an at most countable set is at most countable ($\mathbb{Q}$ is countably infinite, Every subset of an at most countable set is at most countable, Finite, countably infinite, countable, uncountable).

A bounded function on $[a,b]$ with $a < b$ whose set of discontinuities is at most countable is Riemann integrable (A bounded function on $[a,b]$ whose set of discontinuities is at most countable is Riemann integrable, Lower bound, bounded below, bounded set).

The irrationals are dense in $\mathbb{R}$, so every nonempty open interval contains an irrational (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 $P = (n,s)$ of $[0,1]$: $n \ge 1$, $s_i < s_{i+1}$, $\Delta_i > 0$, $I_i = [s_i,s_{i+1}] \subseteq [0,1]$, and $(s_i,s_{i+1})$ is a nonempty open interval (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).

$m_i = \inf t[I_i]$, $L(t,P) = \sum_{i<n}m_i\Delta_i$, $\underline{\int_0^1}t$ is the supremum of the lower sums and $\overline{\int_0^1}t$ the infimum of the upper sums, and the integral is their common value when they 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; the supremum of $\{0\}$ is $0$ (Greatest lower bound (infimum), Maximum and minimum of a set, Complete ordered field (least-upper-bound property)).

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

Ordered-field arithmetic: the order is total and transitive, and a reciprocal of a positive quantity is positive (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.