A continuous function on $[a,b]$ is Riemann integrable, by Heine-Cantor and Riemann's criterion

Given: Reals $a < b$ and a function $f : [a,b] \to \mathbb{R}$ continuous on $[a,b]$.

$[a,b]$ is closed and bounded, hence compact (Intervals of $\mathbb{R}$: the nine order-convex forms, nondegeneracy, and length, Open subset of $\mathbb{R}$ (every point has a neighbourhood inside it), closed subset (complement open), and clopen, Lower bound, bounded below, bounded set, A subset of $\mathbb{R}$ is compact if and only if it is closed and bounded, Open cover, subcover, compact subset of $\mathbb{R}$ (every open cover has a finite subcover), and sequentially compact subset).

A continuous real function on a compact subset of $\mathbb{R}$ is bounded there (A continuous real function on a compact subset of $\mathbb{R}$ is bounded).

Heine-Cantor: a continuous real function on a compact subset $K$ of $\mathbb{R}$ is uniformly continuous on $K$, that is, for every real $\eta > 0$ there is a real $\delta > 0$ with $|f(x) - f(y)| < \eta$ for all $x, y \in K$ with $|x - y| < \delta$ (Heine-Cantor in $\mathbb{R}$: a continuous real function on a compact subset of $\mathbb{R}$ is uniformly continuous, proved $\mathbb{R}$-natively from sequential compactness, Uniform continuity of $f : A \to \mathbb{R}$: one $\delta$ serving every pair of points of $A$).

For a partition $P = (n,t)$ of $[a,b]$: $\Delta_i = t_{i+1} - t_i > 0$, $\sum_{i<n}\Delta_i = b-a$, and the uniform partition $U_N$ into $N \ge 1$ parts has every $\Delta_i$ equal to $(b-a)/\iota(N)$ (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).

$U(f,P) - L(f,P) = \sum_{i<n}(M_i - m_i)\Delta_i$ and $M_i - m_i = \sup\{|f(x)-f(y)| : x, y \in I_i\}$ for bounded $f$ (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 $U(f,P) = \sum_i M_i \Delta_i$, Laws of finite sums and finite products).

Riemann's criterion: a bounded $f$ is integrable if and only if for every real $\varepsilon > 0$ there is a partition $P$ with $U(f,P) - L(f,P) < \varepsilon$ (Riemann's criterion: a bounded $f$ on $[a,b]$ is Darboux integrable if and only if for every real $\varepsilon > 0$ there is a partition $P$ with $U(f,P) - L(f,P) < \varepsilon$).

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).

Finite sums: scaling and monotonicity in the terms (Finite sums and finite products, by recursion, Laws of finite sums and finite products).

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; $x, y \in [c,d]$ gives $|x - y| \le d - c$ (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), Intervals of $\mathbb{R}$: the nine order-convex forms, nondegeneracy, and length). 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.