Proof of the mean value inequality

Builds on Differentiable function of several variables Euclidean space Mean value theorem Chain rule for a composition with a curve Cauchy–Schwarz inequality Operator norm Linear maps on finite-dimensional spaces are continuous Bounded linear maps are continuous Composition of continuous maps is continuous Triangle inequality for norms

Proof

We apply the one-variable mean value theorem to the real function $\varphi(t) = w \cdot f(x + t (y - x))$ , where $w = f(y) - f(x)$ .

Let $U$ , $f$ , $x$ , $y$ and $M$ be as in the statement. Put $w = f(y) - f(x)$ . If $w = 0$ , the inequality holds, since its right side is $\ge 0$ . From now on, suppose that $w \ne 0$ .
Let $z \in U$ and $L = Df(z)$ . By linear maps on finite-dimensional spaces are continuous and bounded linear maps are continuous, there is $C \ge 0$ with $\|L h\| \le C \|h\|$ for each $h$ .
Since $f$ is differentiable at $z$ , there is $\delta > 0$ with $\|f(z + h) - f(z) - L h\| \le \|h\|$ for $\|h\| < \delta$ and $z + h \in U$ . By the triangle inequality, $\|f(z + h) - f(z)\| \le (C + 1) \|h\|$ for these $h$ . So $f$ is continuous at $z$ .
Define $\gamma(t) = x + t (y - x)$ for $t \in [0, 1]$ . By hypothesis $\gamma(t) \in U$ . Also $\|\gamma(t) - \gamma(t')\| = |t - t'| \, \|y - x\|$ , so $\gamma$ is continuous.
Put $\varphi(t) = w \cdot f(\gamma(t))$ for $t \in [0, 1]$ , with the dot product of Euclidean space. By the Cauchy–Schwarz inequality, $|w \cdot v - w \cdot v'| \le \|w\| \, \|v - v'\|$ , so $v \mapsto w \cdot v$ is continuous.
By steps 3, 4 and 5, $\varphi$ is a composition of continuous maps. So $\varphi$ is continuous on $[0, 1]$ .
For $t \in (0, 1)$ and small $s \ne 0$ , $(\gamma(t + s) - \gamma(t)) / s = y - x$ . So $\gamma$ is differentiable on $(0, 1)$ with $\gamma'(t) = y - x$ .
By the chain rule for a composition with a curve on the open interval $(0, 1)$ , $f \circ \gamma$ is differentiable at each $t \in (0, 1)$ . Its derivative is $(f \circ \gamma)'(t) = Df(\gamma(t))(y - x)$ .
For $t \in (0, 1)$ and small $s \ne 0$ , $(\varphi(t + s) - \varphi(t)) / s = w \cdot q(s)$ , where $q(s) = ((f \circ \gamma)(t + s) - (f \circ \gamma)(t)) / s$ . By steps 5 and 8, this tends to $w \cdot Df(\gamma(t))(y - x)$ .
So $\varphi$ is differentiable on $(0, 1)$ with $\varphi'(t) = w \cdot Df(\gamma(t))(y - x)$ .
By the mean value theorem, there is $c \in (0, 1)$ with $\varphi(1) - \varphi(0) = \varphi'(c)$ .
The left side is $w \cdot f(y) - w \cdot f(x) = w \cdot w = \|w\|^2$ . By step 10 and the Cauchy–Schwarz inequality, $\|w\|^2 \le \|w\| \, \|Df(\gamma(c))(y - x)\|$ .
Let $T = Df(\gamma(c))$ and $v = y - x$ . If $v \ne 0$ , then $v / \|v\|$ has norm $1$ . By the definition of the operator norm, $\|T v\| = \|v\| \, \|T(v / \|v\|)\| \le \|T\| \, \|v\|$ . If $v = 0$ , both sides are $0$ .
The point $\gamma(c)$ lies on the segment $[x, y]$ , so $\|T\| \le M$ by hypothesis. Steps 12 and 13 give $\|w\|^2 \le \|w\| \, M \, \|y - x\|$ .
Dividing by $\|w\| > 0$ gives $\|f(y) - f(x)\| \le M \, \|y - x\|$ .

Idea

The mean value theorem has no equality form for vector-valued maps, but it does apply to one real component. The trick is to take the component along the direction $w = f(y) - f(x)$ of the total change. The change of that component is $\|w\|^2$ , and Cauchy–Schwarz bounds its derivative by $\|w\|$ times $M \|y - x\|$ .

Reading path

To understand this page from the ground up, read these 309 pages in order. The foundations come first. The path takes the proofs Proof of the limit of a sum of sequences, Proof of Rolle's theorem, Proof of the approximation property of the supremum, Proof that every sequence has a monotone subsequence, by peaks, Proof that open sets are the sets that are neighbourhoods of all their points, Proof of the commutativity of addition by induction, Proof of the Lebesgue number lemma by a finite cover of balls, Proof of the epsilon-delta form of continuity, Proof of the triangle inequality for norms, Proof that a complex number times its conjugate is the square of its modulus, Proof of the Cauchy–Schwarz inequality, Proof of the Heine–Borel theorem by sequential compactness, Proof that there is a natural number n with 1/n below any positive epsilon, Proof of the chain rule in several variables, Proof of the derivative as a little-o approximation, Proof of the equality of complex numbers, Proof that a linear map between normed spaces is continuous if and only if it is bounded, Proof that a function is bijective iff it has a two-sided inverse, Proof that the reals form an ordered field, with the Cauchy sequence construction, Proof of continuity via closed sets, Proof of the double negation law by a truth table, Proof of the extreme value theorem by the Bolzano–Weierstrass theorem, Proof that compact subsets of a metric space are bounded, Proof that compactness and sequential compactness agree in metric spaces, Proof that subsequences of a convergent sequence converge to the same limit, Proof of the uniqueness of limits, Proof that the coordinate map is a linear isomorphism, Proof of strong induction from ordinary induction, Proof that linear maps on finite-dimensional normed spaces are continuous, Proof of the limit of a product of sequences, Proof that the absolute value is multiplicative, Proof of the squeeze theorem for sequences, Proof of the existence of square roots of nonnegative reals, Proof of the chain rule for a composition with a curve, Proof of the linearity of differentiation, Proof that a continuous image of a compact set is compact, Proof that compact subsets of a Hausdorff space are closed, Proof that the composition of continuous maps is continuous, Proof of epsilon-delta continuity at a point in metric spaces, Proof of the sequential characterization of limits, Proof of the description of the open sets of a metric space, Proof of the triangle inequality for real numbers, Proof of the epsilon-N definition of the limit of a sequence, Proof that compactness of a subset can be tested with open sets of the ambient space, Proof of the existence of the infimum by negation, Proof of the equivalence of norms in finite dimension by compactness of the unit sphere, Proof that every spanning set contains a basis, by Zorn's lemma, Proof of Fermat's theorem on stationary points, Proof that limits preserve non-strict inequalities, Proof that closed sets contain their limits, Proof of Zorn's lemma, Proof that metric spaces are Hausdorff, Proof of the sequence characterisation of uniform continuity, Proof of the epsilon-delta form of the limit of a function, 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 continuous real function on a compact set attains its bounds, 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 that open balls are open, Proof of the boundedness theorem by Bolzano–Weierstrass, Proof that Lipschitz maps are uniformly continuous, Proof that a set is open if and only if its complement is closed, Proof of the mean value theorem from Rolle's theorem, Proof of continuity via open sets in metric spaces, Proof of the constant multiple rule.

Basis concept
Complex numbers object
Complex numbers as pairs of reals concept
Construction of the reals from Cauchy sequences concept
Coordinate space object
Coordinate vector concept
Definition by recursion concept
Element definition
Euclidean metric example
Euclidean space object
Extended real number line object
Function concept
Function composition concept
Imaginary unit object
Inner product concept
Inner product space concept
Integers object
Limit point of a set definition
Linear isomorphism concept
Linear map concept
Local maximum definition
Mathematical induction concept
Natural numbers object
Norm concept
Norm induced by an inner product concept
Ordered field definition
Peano axioms concept
Proof of Zorn's lemma proof
Proposition concept
Rational numbers object
Real numbers object
Sequence definition
Set bounded above definition
Vector space definition
p-metric on R^n example
Real and imaginary parts definition
Set definition
Fréchet derivative definition
Total derivative concept
Function of several variables concept
Vector-valued function concept
Ring of integers statement
Limit of a function definition
Local minimum definition
Successor function definition
Archimedean ordered field 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
Inner product definition
Linear combination definition
Linear map definition
Linear subspace definition
Norm definition
p-norm on R^n example
Complex conjugate definition
Empty set object
Family of sets definition
Ordered pair definition
Differentiable function of several variables concept
Partial derivative concept
Intersection definition
Limit of a function of several variables concept
Union definition
Continuity at a point definition
Derivative definition
Parametrised curve concept
Biconditional definition
Addition of natural numbers definition
Truth table concept
Modus ponens concept
Existential quantifier definition
Universal quantifier definition
Bounded sequence definition
Limit of a sequence definition
Monotone sequence definition
Proof of the Archimedean property in an Archimedean ordered field proof
The rationals form a field statement
The successor function is injective statement
Zero is not a successor statement
Inner product space definition
Linear independence definition
Composition of linear maps definition
Linear isomorphism definition
Linear span definition
Normed vector space definition
Completeness axiom statement
Cartesian product definition
Proof of the equality of complex numbers proof
Gradient concept
Differentiable function of several variables definition
Union of a family of sets definition
Continuous function definition
Subset definition
Differentiable function definition
Proof of the commutativity of addition by induction proof
Mathematical proof concept
Propositional logic concept
Archimedean property statement
Proof of the derivative as a little-o approximation proof
Uniqueness quantifier definition
Order on the natural numbers definition
Convergent sequence definition
Sequential continuity definition
The rationals form an ordered field statement
Norm induced by an inner product definition
Spanning set definition
Equality of complex numbers statement
Binary relation definition
Universal set definition
Topological space definition
Derivative as a little-o approximation statement
Order on the reals definition
Tautology definition
Commutativity of addition statement
Proof of strong induction from ordinary induction proof
Finite-dimensional vector space definition
Basis definition
Antisymmetric relation definition
Reflexive relation definition
Transitive relation definition
Sequential compactness definition
Complement definition
Interval definition
Subspace topology definition
Function definition
Coordinate vector definition
Logical equivalence definition
Strong induction statement
Preorder definition
Choice function definition
Proof of the epsilon-delta form of the limit of a function proof
Image of a set definition
Preimage of a set definition
Codomain definition
Function composition definition
Identity function definition
Injective function definition
Proof of the epsilon-delta form of continuity proof
Metric space definition
Proof of the well-ordering principle from strong induction proof
Proof of the double negation law by a truth table proof
Partial order definition
Axiom of choice statement
Epsilon-delta definition of a limit statement
Surjective function definition
Left inverse definition
Right inverse definition
Epsilon-delta definition of continuity statement
Closed ball definition
Convergence of a sequence in a metric space definition
Lipschitz map definition
Open ball definition
Triangle inequality for a metric statement
Double negation law statement
Diameter of a set definition
Well-ordering principle statement
Maximal element definition
Total order definition
Upper bound definition
Bijective function definition
Metric topology definition
Bounded set definition
Proof by contradiction concept
Proof that every sequence has a monotone subsequence, by peaks proof
Chain definition
Open set definition
Proof that a function is bijective iff it has a two-sided inverse proof
Proof that the reals form an ordered field, with the Cauchy sequence construction proof
Every sequence has a monotone subsequence statement
Hausdorff space definition
Open cover definition
Closed set definition
Continuous map of topological spaces definition
Proof that open sets are the sets that are neighbourhoods of all their points proof
Bijective iff it has a two-sided inverse statement
Equivalent metrics definition
Zorn's lemma statement
Proof of the description of the open sets of a metric space proof
The reals form an ordered field statement
Compact space definition
Proof that a set is open if and only if its complement is closed proof
Open sets characterised by neighbourhoods statement
Proof that the composition of continuous maps is continuous proof
Equivalent norms definition
Open sets of a metric space statement
Proof that the coordinate map is a linear isomorphism proof
Absolute value definition
Proof that every spanning set contains a basis, by Zorn's lemma proof
Proof of the approximation property of the supremum proof
Proof of the existence of the infimum by negation proof
Proof of the existence of square roots of nonnegative reals proof
Complement of an open set is closed statement
Compact set definition
Continuity at a point of a topological space definition
Continuous linear map definition
Composition of continuous maps is continuous statement
Proof that closed sets contain their limits proof
Proof of the triangle inequality for real numbers proof
Proof that the absolute value is multiplicative proof
Proof that open balls are open proof
Every spanning set contains a basis statement
Proof of the epsilon-N definition of the limit of a sequence proof
Isomorphism with the coordinate space statement
Approximation property of the supremum statement
Existence of the infimum statement
Existence of square roots of nonnegative reals statement
Proof that compactness of a subset can be tested with open sets of the ambient space proof
Proof of continuity via closed sets proof
Closed sets contain their limits statement
Triangle inequality statement
Absolute value is multiplicative statement
Open balls are open statement
Square root definition
Epsilon-N definition of the limit of a sequence statement
Proof that the reals are Archimedean from the completeness axiom proof
Compactness via open covers statement
Continuity via closed sets statement
Modulus of a complex number definition
Proof of the limit of a sum of sequences proof
Proof of the uniqueness of limits proof
Proof of epsilon-delta continuity at a point in metric spaces proof
Proof of continuity via open sets in metric spaces proof
Proof of the Lebesgue number lemma by a finite cover of balls proof
Proof that limits preserve non-strict inequalities proof
Proof of the squeeze theorem for sequences proof
Proof that metric spaces are Hausdorff proof
Proof that convergent sequences are bounded proof
The reals are Archimedean statement
Proof that subsequences of a convergent sequence converge to the same limit proof
Proof of the monotone convergence theorem for sequences proof
Proof that compact subsets of a Hausdorff space are closed proof
Proof that a continuous image of a compact set is compact proof
Proof that a complex number times its conjugate is the square of its modulus proof
Lebesgue number lemma statement
Epsilon-delta continuity in metric spaces statement
Proof that compact subsets of a metric space are bounded proof
Metric spaces are Hausdorff statement
Proof that compactness and sequential compactness agree in metric spaces proof
Limit of a sum of sequences statement
Proof of the sequence characterisation of uniform continuity proof
Uniqueness of limits statement
Limits preserve non-strict inequalities statement
Proof of the triangle inequality for norms proof
Convergent sequences are bounded statement
Squeeze theorem statement
Subsequences of a convergent sequence converge to the same limit statement
Proof that there is a natural number n with 1/n below any positive epsilon proof
Monotone convergence theorem for sequences statement
Compact subsets of a Hausdorff space are closed statement
A complex number times its conjugate statement
Continuous image of a compact set is compact statement
Compactness and sequential compactness agree in metric spaces statement
Continuity via open sets statement
Compact subsets of a metric space are bounded statement
Proof of the limit of a product of sequences proof
Epsilon-delta uniform continuity statement
There is a natural number n with 1/n below any positive epsilon statement
Proof that a decreasing sequence bounded below converges proof
Triangle inequality for norms statement
Proof that a linear map between normed spaces is continuous if and only if it is bounded proof
Proof of the sequential characterization of limits proof
Proof that Lipschitz maps are uniformly continuous proof
Proof that continuity is equivalent to sequential continuity proof
Proof of the Cauchy–Schwarz inequality proof
Limit of a product of sequences statement
Decreasing sequence bounded below converges statement
Lipschitz maps are uniformly continuous statement
Continuity is equivalent to sequential continuity statement
Sequential characterization of limits statement
Proof of the Bolzano–Weierstrass theorem by a monotone subsequence proof
Cauchy–Schwarz inequality statement
Bounded linear maps are continuous statement
Proof of the linearity of differentiation proof
Bolzano–Weierstrass theorem statement
Linearity of differentiation statement
Proof of the Heine–Borel theorem by sequential compactness proof
Operator norm definition
Proof of the boundedness theorem by Bolzano–Weierstrass proof
Proof of the constant multiple rule proof
Boundedness theorem statement
Heine–Borel theorem statement
Constant multiple rule statement
Proof of the extreme value theorem by the Bolzano–Weierstrass theorem proof
Proof that a continuous real function on a compact set attains its bounds proof
Extreme value theorem statement
Continuous real function on a compact set attains its bounds statement
Proof of Fermat's theorem on stationary points proof
Proof of the equivalence of norms in finite dimension by compactness of the unit sphere proof
Fermat's theorem on stationary points statement
Equivalence of norms in finite dimension statement
Proof of Rolle's theorem proof
Rolle's theorem statement
Proof that linear maps on finite-dimensional normed spaces are continuous proof
Linear maps on finite-dimensional spaces are continuous statement
Proof of the mean value theorem from Rolle's theorem proof
Mean value theorem statement
Proof of the chain rule in several variables proof
Chain rule in several variables statement
Proof of the chain rule for a composition with a curve proof
Chain rule for a composition with a curve statement
Proof of the mean value inequality proof

Where this leads

282 pages build on this one.

Tag	`ZKWK0NE4J8`
URI	`m2://ZKWK0NE4J8`
Kind	proof
Writing	W0 stub
MSC	26B05
Proves	Mean value inequality
Uses	Differentiable function of several variables, Euclidean space, Mean value theorem, Chain rule for a composition with a curve, Cauchy–Schwarz inequality, Operator norm, Linear maps on finite-dimensional spaces are continuous, Bounded linear maps are continuous, Composition of continuous maps is continuous, Triangle inequality for norms
Changed	4 November 2026

Proof of the mean value inequality

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