theorem 7.100 $C^{1}$ implies differentiable

open in the book · parts/02-mathematical-methods/05-real-analysis.tex:2957 · p. 242

Rests on

depends_on definition 7.98 Functions of class $C^{1}$ ¶
- depends_on definition 7.20 Continuity at a point ¶
  - depends_on definition 7.16 Limit ¶
    - depends_on definition 7.2 Absolute value ¶
    - depends_on definition 7.9 Real function ¶
      - depends_on definition 3.43 Map ¶
  - depends_on equation 7.7 eq:ana-limit-left ¶
  - depends_on equation 7.5 eq:ana-limit-right ¶
- depends_on definition 7.97 Partial derivative; gradient ¶
  - depends_on definition 7.26 Derivative of a function at a point ¶
    - depends_on definition 7.16 Limit ¶ ↺
    - depends_on definition 7.9 Real function ¶ ↺
  - depends_on definition 5.15 Basis ¶
    - depends_on definition 5.12 Subspace generated by a set of vectors ¶
      - depends_on definition 5.5 Linear combination ¶
      - depends_on definition 5.7 Vector subspace ¶
      - proves proof ch:03-linear-algebra-representations@proof-3 ¶
    - depends_on definition 5.14 Linear independence ¶
      - depends_on definition 5.5 Linear combination ¶ ↺
depends_on definition 7.99 Differentiability at a point ¶
- depends_on definition 7.16 Limit ¶ ↺
- depends_on definition 5.37 Linear transformation ¶
  - depends_on definition 4.33 Vector space ¶
    - depends_on definition 4.8 Commutativity; abelian structure ¶
      - depends_on definition 4.4 Internal binary operation; magma ¶
    - depends_on definition 4.32 Field ¶
      - depends_on definition 4.11 Inverse element ¶
      - depends_on definition 4.20 Monoid ¶
      - depends_on definition 4.30 Ring ¶
    - depends_on definition 4.31 Module ¶
      - depends_on definition 4.29 Action ¶
      - depends_on definition 4.28 Distributivity ¶
      - depends_on definition 4.21 Group ¶
      - depends_on definition 4.30 Ring ¶ ↺
- depends_on equation 6.12 eq:top-euclidean-metric ¶
depends_on theorem 7.35 Mean value theorem ¶
- depends_on proposition 7.29 Linearity ¶
  - depends_on definition 7.11 Sum of functions ¶
    - depends_on definition 7.9 Real function ¶ ↺
  - depends_on equation 7.14 eq:ana-derivh ¶
  - depends_on proposition 7.6 Algebra of limits ¶
    - depends_on definition 7.4 Convergence ¶
      - depends_on definition 7.2 Absolute value ¶ ↺
    - depends_on proposition 7.3 Triangle inequality ¶
      - depends_on definition 7.2 Absolute value ¶ ↺
      - proves proof ch:05-real-analysis@proof-1 ¶
    - proves proof ch:05-real-analysis@proof-3 ¶
  - proves proof ch:05-real-analysis@proof-13 ¶
- depends_on theorem 7.34 Rolle ¶
  - depends_on lemma 7.33 Fermat: interior extremum ¶
    - depends_on definition 7.26 Derivative of a function at a point ¶ ↺
    - depends_on proposition 7.18 Two-sided limit from one-sided limits ¶
      - depends_on definition 7.16 Limit ¶ ↺
      - depends_on equation 7.7 eq:ana-limit-left ¶ ↺
      - depends_on equation 7.5 eq:ana-limit-right ¶ ↺
      - proves proof ch:05-real-analysis@proof-6 ¶
    - proves proof ch:05-real-analysis@proof-17 ¶
  - depends_on theorem 7.24 Extreme value theorem ¶
    - depends_on axiom 7.1 Completeness of $\R$ ¶
    - depends_on proposition 7.22 Sequential characterization ¶
      - depends_on definition 7.20 Continuity at a point ¶ ↺
      - depends_on definition 7.4 Convergence ¶ ↺
      - proves proof ch:05-real-analysis@proof-7 ¶
    - depends_on theorem 7.7 Bolzano–Weierstrass ¶
      - depends_on corollary A.45 Archimedean property and density of $\Q$ ¶
      - depends_on corollary A.46 Monotone convergence ¶
      - proves proof ch:05-real-analysis@proof-4 ¶
    - proves proof ch:05-real-analysis@proof-9 ¶
  - proves proof ch:05-real-analysis@proof-18 ¶
- proves proof ch:05-real-analysis@proof-19 ¶
proves proof ch:05-real-analysis@proof-62 ¶

Supports

depends_on lemma A.291 $h$ is differentiable, with the stated derivative ¶
depends_on remark 7.101 Partial derivatives alone do not suffice ¶
depends_on theorem 7.112 Implicit function theorem ¶
- depends_on corollary 7.113 Inverse function theorem ¶
  - depends_on definition A.503 The substitution property ¶
    - depends_on lemma A.506 Locality ¶
      - depends_on proposition A.511 The substitution property is universal ¶
    - depends_on lemma A.504 Transitivity ¶
  - depends_on lemma 7.116 Functions vanishing on a regular zero set ¶
  - depends_on lemma A.510 Every diffeomorphism factorises locally ¶
    - depends_on proposition A.511 The substitution property is universal ¶ ↺
  - depends_on remark 7.114 Where the proof is, and why ¶
  - depends_on theorem 7.129 Change of variables in a multiple integral ¶
    - depends_on example 7.130 The two Jacobians this treatise uses ¶
      - depends_on lemma A.518 The excised ball ¶
    - depends_on lemma A.518 The excised ball ¶ ↺
    - depends_on proposition A.519 The convolution exists ¶
      - depends_on proposition A.524 Rate of decay ¶
  - depends_on theorem 7.115 Constant rank ¶
- depends_on proposition 7.120 The envelope touches every member it meets ¶
- depends_on proposition 7.117 Lagrange multipliers in finitely many variables ¶
  - depends_on remark 7.118 The multiplier rule and its functional counterpart ¶
- depends_on remark 7.114 Where the proof is, and why ¶ ↺
depends_on theorem A.286 Implicit function theorem ¶
- depends_on corollary A.292 Inverse function theorem ¶
  - depends_on lemma A.549 The action is transitive ¶
    - depends_on proposition A.552 The component is a torus ¶
      - depends_on corollary A.553 Quasi-periodic motion ¶
      - depends_on definition A.554 Actions on the torus ¶
      - depends_on proposition A.556 The Jacobian of the actions is the lattice matrix ¶
  - depends_on lemma A.300 The boundary is well defined, and is a manifold ¶
    - depends_on definition A.309 The induced orientation of the boundary ¶
      - depends_on theorem A.311 General Stokes theorem ¶
  - depends_on proposition A.535 Existence of a slice ¶
    - depends_on lemma A.536 A slice is a chart domain downstairs ¶
      - depends_on theorem A.537 The smooth structure on the orbit space ¶
    - depends_on remark A.542 Where each hypothesis is spent ¶
  - depends_on proposition 22.32 Duality of the two brackets ¶
    - depends_on remark 22.33 What the duality is good for ¶
  - depends_on proposition 22.3 Invertibility of the Legendre map ¶
    - depends_on definition 26.3 Singular Lagrangian ¶
      - depends_on proposition 26.5 Primary constraints ¶
    - depends_on remark 22.4 When the Legendre map is singular ¶
  - depends_on proposition 13.132 Simultaneous straightening of commuting fields ¶
    - depends_on theorem 13.133 Frobenius ¶
      - depends_on corollary 13.134 Frobenius for a Pfaffian system ¶
      - depends_on proposition A.547 The components of the level set are the leaves of an integrable distribution ¶
      - depends_on theorem A.545 Liouville–Arnold ¶
    - depends_on theorem 13.137 Commuting complete fields on a compact manifold ¶
  - depends_on remark 22.19 Which form exists ¶
  - depends_on theorem 13.63 Constant rank theorem ¶
    - depends_on corollary 13.64 The image of a constant-rank map, locally ¶
    - depends_on lemma A.533 The orbit map has constant rank $d$ ¶
      - depends_on proposition A.534 Every orbit is an embedded copy of $G$ ¶
      - depends_on proposition A.535 Existence of a slice ¶ ↺
    - depends_on lemma A.538 Submersions have smooth local sections ¶
      - depends_on proposition A.539 Universal property ¶
    - depends_on theorem 13.67 Quotient manifold theorem ¶
      - depends_on example 13.68 Why each hypothesis is there ¶
      - depends_on proposition A.566 The isotropy group acts, and the quotient is smooth ¶
      - depends_on theorem A.563 Marsden–Weinstein reduction ¶
- depends_on corollary A.293 Solving one scalar equation for one coordinate ¶
  - depends_on example A.294 The sphere, made explicit ¶
- depends_on lemma A.650 Flattening the constraints ¶
  - depends_on lemma A.651 Acyclicity of $\delta$ ¶
    - depends_on proposition A.655 Uniqueness up to a canonical transformation ¶
    - depends_on theorem A.653 Existence of a nilpotent BRST charge ¶
      - depends_on proposition A.655 Uniqueness up to a canonical transformation ¶ ↺
- depends_on remark A.296 Both hypotheses are needed, and the conclusion is local ¶
- depends_on remark 23.3 What Part II owes this chapter ¶
- depends_on theorem 23.6 Jacobi ¶
  - depends_on example 23.25 Central force in plane polar coordinates ¶
  - depends_on example 23.22 Free particle ¶
    - depends_on remark 23.42 Phase speed and particle speed ¶
  - depends_on example 23.23 Uniform gravity, and the parabola ¶
  - depends_on proposition 23.7 The principal function is the action ¶
    - depends_on remark 23.8 The action as a function, not a functional ¶
- depends_on theorem 13.63 Constant rank theorem ¶ ↺
- depends_on theorem 13.62 Regular value theorem in codimension $k$ ¶
  - depends_on corollary 13.64 The image of a constant-rank map, locally ¶ ↺

Neighborhood

Every logical edge within two steps of this node.

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partly declared
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Edges

type	direction	node	provenance	where
`depends_on`	→	Functions of class $C^{1}$	declared	parts/02-mathematical-methods/05-real-analysis.tex:2967
`depends_on`	→	Differentiability at a point	declared	parts/02-mathematical-methods/05-real-analysis.tex:2967
`depends_on`	→	Mean value theorem	declared	parts/02-mathematical-methods/05-real-analysis.tex:2967
`depends_on`	←	$h$ is differentiable, with the stated derivative	declared	appendices/A-long-proofs.tex:14587
`depends_on`	←	Partial derivatives alone do not suffice	declared	parts/02-mathematical-methods/05-real-analysis.tex:3062
`depends_on`	←	Implicit function theorem	declared	parts/02-mathematical-methods/05-real-analysis.tex:3523
`depends_on`	←	Implicit function theorem	declared	appendices/A-long-proofs.tex:14371
`proves`	←	ch:05-real-analysis@proof-62	declared	parts/02-mathematical-methods/05-real-analysis.tex:2970