proposition 7.120 The envelope touches every member it meets

open in the book · parts/02-mathematical-methods/05-real-analysis.tex:3826 · p. 251

Rests on

depends_on definition 7.119 Envelope of a family ¶
- 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 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 5.14 Linear independence ¶
- depends_on definition 7.97 Partial derivative; gradient ¶ ↺
depends_on proposition 7.104 Chain rule in several variables ¶
- depends_on definition 7.13 Composite function ¶
  - depends_on definition 7.9 Real function ¶ ↺
- 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 4.32 Field ¶
      - depends_on definition 4.31 Module ¶
  - depends_on equation 6.12 eq:top-euclidean-metric ¶
- depends_on definition 7.102 Differential of a function ¶
  - depends_on definition 7.99 Differentiability at a point ¶ ↺
  - depends_on definition 7.97 Partial derivative; gradient ¶ ↺
- proves proof ch:05-real-analysis@proof-63 ¶
depends_on theorem 7.112 Implicit function theorem ¶
- depends_on definition 7.99 Differentiability at a point ¶ ↺
- depends_on proposition 7.104 Chain rule in several variables ¶ ↺
- 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 ¶
    - depends_on proposition 7.29 Linearity ¶
      - depends_on definition 7.11 Sum of functions ¶
      - depends_on equation 7.14 eq:ana-derivh ¶
      - depends_on proposition 7.6 Algebra of limits ¶
      - proves proof ch:05-real-analysis@proof-13 ¶
    - depends_on theorem 7.34 Rolle ¶
      - depends_on lemma 7.33 Fermat: interior extremum ¶
      - depends_on theorem 7.24 Extreme value theorem ¶
      - proves proof ch:05-real-analysis@proof-18 ¶
    - proves proof ch:05-real-analysis@proof-19 ¶
  - proves proof ch:05-real-analysis@proof-62 ¶
- proves proof ch:05-real-analysis@prooflink-1 ¶
proves proof ch:05-real-analysis@proof-73 ¶

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Edges

type	direction	node	provenance	where
`depends_on`	→	Envelope of a family	declared	parts/02-mathematical-methods/05-real-analysis.tex:3840
`depends_on`	→	Chain rule in several variables	declared	parts/02-mathematical-methods/05-real-analysis.tex:3840
`depends_on`	→	Implicit function theorem	declared	parts/02-mathematical-methods/05-real-analysis.tex:3840
`proves`	←	ch:05-real-analysis@proof-73	declared	parts/02-mathematical-methods/05-real-analysis.tex:3844