Proof that continuous functions are Riemann integrable

Builds on Boundedness theorem Continuous functions on a closed bounded interval are uniformly continuous Uniform continuity on a set The reals are Archimedean Extreme value theorem Upper and lower Darboux sums Riemann's criterion for integrability

Proof

We use uniform continuity and Riemann's criterion.

Let $a < b$ and let $f \colon [a, b] \to \mathbb{R}$ be continuous. By the boundedness theorem, $f$ is bounded. So its Darboux sums are defined.
Let $\varepsilon > 0$ . By the uniform continuity theorem, $f$ is uniformly continuous on $[a, b]$ . So there is $\delta > 0$ with $|f(x) - f(y)| < \frac{\varepsilon}{2(b - a)}$ for all $x, y \in [a, b]$ with $|x - y| < \delta$ .
By the Archimedean property, take $n$ with $n \delta > b - a$ . Let $P$ have the points $x_i = a + i (b - a)/n$ for $i = 0, \dots, n$ . Each subinterval has length $(b - a)/n < \delta$ .
Fix $i$ . The restriction of $f$ to $[x_{i-1}, x_i]$ is continuous. By the extreme value theorem, there are $c_i, d_i \in [x_{i-1}, x_i]$ with $f(c_i) \le f(x) \le f(d_i)$ for all $x$ there.
So $M_i = f(d_i)$ and $m_i = f(c_i)$ . Since $|d_i - c_i| \le (b - a)/n < \delta$ , step 2 gives $M_i - m_i < \frac{\varepsilon}{2(b - a)}$ .
Hence $U(f, P) - L(f, P) = \sum_{i=1}^{n} (M_i - m_i)(x_i - x_{i-1}) < \frac{\varepsilon}{2(b - a)} \sum_{i=1}^{n} (x_i - x_{i-1}) = \frac{\varepsilon}{2} < \varepsilon.$
Such a partition exists for each $\varepsilon > 0$ . By Riemann's criterion, $f$ is Riemann integrable on $[a, b]$ .

Idea

Uniform continuity gives one $\delta$ that works on the whole interval. On every subinterval shorter than $\delta$ , the largest and smallest values of $f$ differ by a tiny amount. So the upper and lower sums differ by that amount times $b - a$ .

Reading path

To understand this page from the ground up, read these 130 pages in order. The foundations come first. The path takes the proofs Proof that Darboux and Riemann integrals agree, Proof of the limit of a sum of sequences, Proof of the approximation property of the supremum, Proof that every sequence has a monotone subsequence, by peaks, Proof of the commutativity of addition by induction, Proof of the epsilon-delta form of continuity, Proof that there is a natural number n with 1/n below any positive epsilon, Proof that continuous functions on a closed bounded interval are uniformly continuous, Proof that the reals form an ordered field, with the Cauchy sequence construction, Proof of the double negation law by a truth table, Proof of the extreme value theorem by the Bolzano–Weierstrass theorem, Proof of the uniqueness of limits, Proof of strong induction from ordinary induction, Proof of the squeeze theorem for sequences, Proof of Riemann's criterion for integrability, Proof of the triangle inequality for real numbers, Proof of the epsilon-N definition of the limit of a sequence, Proof that limits preserve non-strict inequalities, Proof of the well-ordering principle from strong induction, Proof of the Bolzano–Weierstrass theorem by a monotone subsequence, Proof of the Archimedean property in an Archimedean ordered field, Proof that a decreasing sequence bounded below converges, Proof that the reals are Archimedean from the completeness axiom, Proof that convergent sequences are bounded, Proof of the monotone convergence theorem for sequences, Proof that continuity is equivalent to sequential continuity, Proof of the boundedness theorem by Bolzano–Weierstrass.

Construction of the reals from Cauchy sequences concept
Definition by recursion concept
Element definition
Integers object
Limit point of a set definition
Mathematical induction concept
Natural numbers object
Ordered field definition
Partition of an interval definition
Peano axioms concept
Proposition concept
Rational numbers object
Real numbers object
Sequence definition
Set bounded above definition
Set definition
Ring of integers statement
Limit of a function definition
Archimedean ordered field definition
Successor function definition
Mesh of a partition definition
Refinement of a partition definition
Riemann sum definition
Conjunction definition
Disjunction definition
Implication definition
Negation definition
Predicate definition
The reals form a field statement
Real sequence definition
Subsequence definition
Infimum definition
Supremum definition
Ordered pair definition
Continuity at a point definition
Addition of natural numbers definition
Proof of the Archimedean property in an Archimedean ordered field proof
Riemann integrable function definition
The rationals form a field statement
The successor function is injective statement
Zero is not a successor statement
Biconditional definition
Truth table concept
Modus ponens concept
Universal quantifier definition
Bounded sequence definition
Limit of a sequence definition
Monotone sequence definition
Completeness axiom statement
Cartesian product definition
Continuous function definition
Archimedean property statement
Proof of the commutativity of addition by induction proof
Order on the natural numbers definition
Riemann integral definition
Subset definition
The rationals form an ordered field statement
Propositional logic concept
Mathematical proof concept
Upper and lower Darboux sums definition
Convergent sequence definition
Sequential continuity definition
Binary relation definition
Commutativity of addition statement
Order on the reals definition
Proof of strong induction from ordinary induction proof
Tautology definition
Upper and lower Darboux integrals definition
Antisymmetric relation definition
Reflexive relation definition
Transitive relation definition
Interval definition
Strong induction statement
Logical equivalence definition
Preorder definition
Proof of the epsilon-delta form of continuity proof
Proof of the well-ordering principle from strong induction proof
Proof of the double negation law by a truth table proof
Partial order definition
Epsilon-delta definition of continuity statement
Well-ordering principle statement
Double negation law statement
Total order definition
Uniform continuity definition
Proof by contradiction concept
Proof that every sequence has a monotone subsequence, by peaks proof
Uniform continuity on a set definition
Proof that the reals form an ordered field, with the Cauchy sequence construction proof
Every sequence has a monotone subsequence statement
The reals form an ordered field statement
Absolute value definition
Proof of the approximation property of the supremum proof
Proof of the triangle inequality for real numbers proof
Proof of the epsilon-N definition of the limit of a sequence proof
Approximation property of the supremum statement
Triangle inequality statement
Epsilon-N definition of the limit of a sequence statement
Proof that the reals are Archimedean from the completeness axiom proof
Proof of the limit of a sum of sequences proof
Proof of the squeeze theorem for sequences proof
Proof of the uniqueness of limits proof
Proof that limits preserve non-strict inequalities proof
The reals are Archimedean statement
Proof that convergent sequences are bounded proof
Proof of the monotone convergence theorem for sequences proof
Proof that there is a natural number n with 1/n below any positive epsilon proof
Limit of a sum of sequences statement
Uniqueness of limits statement
Limits preserve non-strict inequalities statement
Squeeze theorem statement
Convergent sequences are bounded statement
Proof that Darboux and Riemann integrals agree proof
Monotone convergence theorem for sequences statement
There is a natural number n with 1/n below any positive epsilon statement
Darboux and Riemann integrals agree statement
Proof that a decreasing sequence bounded below converges proof
Decreasing sequence bounded below converges statement
Proof that continuity is equivalent to sequential continuity proof
Proof of Riemann's criterion for integrability proof
Continuity is equivalent to sequential continuity statement
Riemann's criterion for integrability statement
Proof of the Bolzano–Weierstrass theorem by a monotone subsequence proof
Bolzano–Weierstrass theorem statement
Proof that continuous functions on a closed bounded interval are uniformly continuous proof
Proof of the boundedness theorem by Bolzano–Weierstrass proof
Continuous functions on a closed bounded interval are uniformly continuous statement
Boundedness theorem statement
Proof of the extreme value theorem by the Bolzano–Weierstrass theorem proof
Extreme value theorem statement
Proof that continuous functions are Riemann integrable proof

Where this leads

695 pages build on this one.

Tag	`ZJZ2Q4Z3C1`
URI	`m2://ZJZ2Q4Z3C1`
Kind	proof
Writing	W0 stub
MSC	26A42
Proves	Continuous functions are Riemann integrable
Uses	Boundedness theorem, Continuous functions on a closed bounded interval are uniformly continuous, Uniform continuity on a set, The reals are Archimedean, Extreme value theorem, Upper and lower Darboux sums, Riemann's criterion for integrability
Changed	4 November 2026

Proof that continuous functions are Riemann integrable

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