Proof of the mean value theorem for integrals

Builds on Continuous functions are Riemann integrable Extreme value theorem Riemann sum Riemann integrable function Monotonicity of the integral Intermediate value theorem

Proof

We trap the average value of $f$ between its minimum and maximum and apply the intermediate value theorem.

Let $a < b$ and let $f \colon [a, b] \to \mathbb{R}$ be continuous. Since continuous functions are Riemann integrable, the integral $I = \int_a^b f(x) \, dx$ exists.
By the extreme value theorem, there are $p, q \in [a, b]$ with $m = f(p) \le f(x) \le f(q) = M$ for all $x \in [a, b]$ .
Let $k$ be a real constant. Every Riemann sum of the constant function $k$ is $\sum_{i} k (x_i - x_{i-1}) = k (b - a)$ . So the constant $k$ is Riemann integrable with integral $k (b - a)$ .
By step 2, $m \le f \le M$ on $[a, b]$ . By the monotonicity of the integral and step 3, $m (b - a) \le I \le M (b - a)$ .
Let $\mu = \frac{I}{b - a}$ . Dividing step 4 by $b - a > 0$ gives $f(p) = m \le \mu \le M = f(q)$ .
If $p = q$ , then $m = M = \mu$ , and $c = p$ satisfies $f(c) = \mu$ .
If $p \ne q$ , let $[r, s]$ be the closed interval with endpoints $p$ and $q$ . The restriction of $f$ to $[r, s]$ is continuous, and $\mu$ lies between $f(r)$ and $f(s)$ . By the intermediate value theorem, there is $c \in [r, s] \subseteq [a, b]$ with $f(c) = \mu$ .
In both cases $c \in [a, b]$ and $\int_a^b f(x) \, dx = \mu (b - a) = f(c)(b - a)$ .

Idea

The average value $\frac{1}{b - a} \int_a^b f$ lies between the smallest and the largest value of $f$ . A continuous function on an interval takes every value between two of its values. So it takes its average value somewhere.

Reading path

To understand this page from the ground up, read these 137 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 intermediate value theorem by the supremum, 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 monotonicity of the integral, Proof of the boundedness theorem by Bolzano–Weierstrass, Proof that continuous functions are Riemann integrable.

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
Definite integral definition
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 of the intermediate value theorem by the supremum proof
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
Intermediate value theorem statement
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
Proof of the monotonicity of the integral proof
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
Monotonicity of the integral statement
Darboux and Riemann integrals agree statement
Proof that a decreasing sequence bounded below converges proof
Proof that continuity is equivalent to sequential continuity proof
Decreasing sequence bounded below converges statement
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
Continuous functions are Riemann integrable statement
Proof of the mean value theorem for integrals proof

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626 pages build on this one.

Tag	`F89W27S5QT`
URI	`m2://F89W27S5QT`
Kind	proof
Writing	W0 stub
MSC	26A42
Proves	Mean value theorem for integrals
Uses	Continuous functions are Riemann integrable, Extreme value theorem, Riemann sum, Riemann integrable function, Monotonicity of the integral, Intermediate value theorem
Changed	4 November 2026

Proof of the mean value theorem for integrals

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