lemma A.291 $h$ is differentiable, with the stated derivative

open in the book · appendices/A-long-proofs.tex:14581 · p. 2936

Rests on

depends_on definition 7.99 Differentiability 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 definition 3.24 Quantifiers ¶
      - depends_on definition 3.28 Set ¶
- 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 lemma A.290 $h$ is Lipschitz ¶
- depends_on lemma A.288 The iteration converges ¶
  - depends_on lemma A.287 The Newton map contracts ¶
    - depends_on definition 7.98 Functions of class $C^{1}$ ¶
      - depends_on definition 7.20 Continuity at a point ¶
      - depends_on definition 7.97 Partial derivative; gradient ¶
    - depends_on equation A.523 eq:app-implicit-hypothesis ¶
    - depends_on theorem 7.35 Mean value theorem ¶
      - depends_on proposition 7.29 Linearity ¶
      - depends_on theorem 7.34 Rolle ¶
      - proves proof ch:05-real-analysis@proof-19 ¶
    - proves proof app:A-long-proofs@proof-180 ¶
  - depends_on proposition 7.46 Geometric series ¶
    - depends_on corollary A.46 Monotone convergence ¶
      - depends_on theorem A.40 Least-upper-bound property ¶
      - proves proof app:A-long-proofs@proof-33 ¶
    - depends_on definition 7.45 Series ¶
      - depends_on definition 7.2 Absolute value ¶ ↺
      - depends_on definition 7.4 Convergence ¶
    - proves proof ch:05-real-analysis@proof-27 ¶
  - depends_on theorem 7.8 Cauchy criterion ¶
    - depends_on corollary A.47 Cauchy completeness ¶
      - depends_on corollary A.46 Monotone convergence ¶ ↺
      - depends_on theorem A.40 Least-upper-bound property ¶ ↺
      - proves proof app:A-long-proofs@proof-34 ¶
    - depends_on definition 7.4 Convergence ¶ ↺
    - proves proof ch:05-real-analysis@proof-5 ¶
  - proves proof app:A-long-proofs@proof-181 ¶
- depends_on theorem 7.35 Mean value theorem ¶ ↺
- proves proof app:A-long-proofs@proof-183 ¶
depends_on theorem 7.100 $C^{1}$ implies differentiable ¶
- depends_on definition 7.98 Functions of class $C^{1}$ ¶ ↺
- depends_on definition 7.99 Differentiability at a point ¶ ↺
- depends_on theorem 7.35 Mean value theorem ¶ ↺
- proves proof ch:05-real-analysis@proof-62 ¶
proves proof app:A-long-proofs@proof-184 ¶

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Edges

type	direction	node	provenance	where
`depends_on`	→	Differentiability at a point	declared	appendices/A-long-proofs.tex:14587
`depends_on`	→	$h$ is Lipschitz	declared	appendices/A-long-proofs.tex:14587
`depends_on`	→	$C^{1}$ implies differentiable	declared	appendices/A-long-proofs.tex:14587
`proves`	←	app:A-long-proofs@proof-184	declared	appendices/A-long-proofs.tex:14591