Proof that C^1 functions are strictly differentiable

Builds on Strict differentiability C^k function characterised by continuous derivatives Continuous partial derivatives imply differentiability Jacobian matrix Operator norm Cauchy–Schwarz inequality Mean value inequality Convex set

Proof

We apply the mean value inequality to $f$ minus its linear part at $a$ .

Let $U$ and $f$ be as in the statement, and let $a \in U$ . Write $\norm{B}$ for the operator norm of a matrix $B$ .
Since $f$ is of class C^1, it is differentiable at each point $z \in U$ , and $Df(z)$ is the Jacobian matrix of partial derivatives at $z$ .
For a matrix $B = (b_{ij})$ , the Cauchy–Schwarz inequality on each row gives $\norm{B} \le (\sum_{i, j} b_{ij}^2)^{1/2}$ .
Let $\varepsilon > 0$ . The partial derivatives of $f$ are continuous at $a$ . By step 3, there is $r > 0$ such that the ball $B_r = \set{z : \norm{z - a} < r}$ lies in $U$ and $\norm{Df(z) - Df(a)} \le \varepsilon$ for $z \in B_r$ .
Let $g(z) = f(z) - Df(a) z$ for $z \in U$ . A linear map is its own derivative, so $g$ is differentiable on $U$ with $Dg(z) = Df(z) - Df(a)$ .
The ball $B_r$ is convex, so it contains the segment between any two of its points. By the mean value inequality and step 4, $\norm{g(x) - g(y)} \le \varepsilon \norm{x - y}$ for $x, y \in B_r$ .
Since $g(x) - g(y) = f(x) - f(y) - Df(a)(x - y)$ , this says $\norm{f(x) - f(y) - Df(a)(x - y)} \le \varepsilon \norm{x - y}$ for $x, y \in B_r$ .
If $\norm{(x, y) - (a, a)} < r$ , then $\norm{x - a} < r$ and $\norm{y - a} < r$ , so $x, y \in B_r$ . For such pairs with $x \ne y$ , step 7 gives $\frac{\norm{f(x) - f(y) - Df(a)(x - y)}}{\norm{x - y}} \le \varepsilon$ .
Since $\varepsilon > 0$ was arbitrary, this quotient tends to $0$ as $(x, y) \to (a, a)$ with $x \ne y$ . So $f$ is strictly differentiable at $a$ , with linear map $Df(a)$ .

Idea

Subtracting the linear map $Df(a)$ leaves a function whose derivative is small near $a$ , because $Df$ is continuous. The mean value inequality turns a small derivative into a small Lipschitz constant, and this is exactly the estimate that strict differentiability asks for.

Reading path

To understand this page from the ground up, read these 334 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 derivative of a map into R^n by components, 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 that the derivative gives the directional derivatives, Proof that differentiable functions are continuous, 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 of the derivative of a map into a product, 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 C^k functions are characterised by continuous derivatives, Proof that continuous partial derivatives imply differentiability, 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 mean value inequality, Proof of the constant multiple rule.

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Proof that the derivative gives the directional derivatives proof
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Zorn's lemma statement
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Proof that the composition of continuous maps is continuous proof
Proof that a set is open if and only if its complement is closed proof
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Proof that the coordinate map is a linear isomorphism proof
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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
Compact set definition
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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
Proof of the epsilon-N definition of the limit of a sequence proof
Every spanning set contains a basis statement
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
Epsilon-N definition of the limit of a sequence statement
Square root definition
Proof that the reals are Archimedean from the completeness axiom proof
Compactness via open covers statement
Modulus of a complex number definition
Continuity via closed sets statement
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 the Lebesgue number lemma by a finite cover of balls proof
Proof of continuity via open sets in metric spaces proof
Proof that limits preserve non-strict inequalities proof
Proof of the squeeze theorem for sequences proof
Proof that differentiable functions are continuous 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 complex number times its conjugate is the square of its modulus proof
Proof that a continuous image of a compact set is compact proof
Lebesgue number lemma statement
Proof that compact subsets of a metric space are bounded proof
Epsilon-delta continuity in metric spaces statement
Proof that compactness and sequential compactness agree in metric spaces proof
Metric spaces are Hausdorff statement
Limit of a sum of sequences statement
Differentiable functions are continuous statement
Uniqueness of limits statement
Proof of the sequence characterisation of uniform continuity proof
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
A complex number times its conjugate statement
Compact subsets of a Hausdorff space are closed statement
Continuous image of a compact set is compact statement
Compactness and sequential compactness agree in metric spaces statement
Compact subsets of a metric space are bounded statement
Continuity via open sets 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 that Lipschitz maps are uniformly continuous proof
Proof of the sequential characterization of limits 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
Proof of the derivative of a map into a product proof
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Operator norm definition
Proof of the boundedness theorem by Bolzano–Weierstrass proof
Proof of the constant multiple rule proof
Derivative of a map into a product statement
Heine–Borel theorem statement
Boundedness theorem statement
Constant multiple rule statement
Derivative of a map into R^n by components 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
Proof that continuous partial derivatives imply differentiability proof
Chain rule in several variables statement
Continuous partial derivatives imply differentiability statement
Proof of the chain rule for a composition with a curve proof
Chain rule for a composition with a curve statement
Proof that C^k functions are characterised by continuous derivatives proof
C^k function characterised by continuous derivatives statement
Proof of the mean value inequality proof
Mean value inequality statement
Proof that C^1 functions are strictly differentiable proof

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1 page builds on this one.

C^1 functions are strictly differentiable

Tag	`HDMDRA7ZBW`
URI	`m2://HDMDRA7ZBW`
Kind	proof
Writing	W0 stub
MSC	26B05
Proves	C^1 functions are strictly differentiable
Uses	Strict differentiability, C^k function characterised by continuous derivatives, Continuous partial derivatives imply differentiability, Jacobian matrix, Operator norm, Cauchy–Schwarz inequality, Mean value inequality, Convex set
Changed	4 November 2026

Proof that C^1 functions are strictly differentiable

Proof

Idea

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