lemma 6.19 Continuous argument along a path

open in the book · parts/02-mathematical-methods/04-topology.tex:309 · p. 197

Rests on

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 7.16 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 7.4 Convergence ¶
    - depends_on definition 7.2 Absolute value ¶ ↺
  - 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 definition A.36 Cut ¶
    - depends_on theorem A.40 Least-upper-bound property ¶
      - depends_on definition A.36 Cut ¶ ↺
      - depends_on definition A.38 Order on $\R$ ¶
      - proves proof app:A-long-proofs@proof-29 ¶
    - proves proof app:A-long-proofs@proof-32 ¶
  - depends_on corollary A.46 Monotone convergence ¶
    - depends_on theorem A.40 Least-upper-bound property ¶ ↺
    - proves proof app:A-long-proofs@proof-33 ¶
  - proves proof ch:05-real-analysis@proof-4 ¶
- proves proof ch:05-real-analysis@proof-9 ¶
depends_on theorem 7.25 Heine–Cantor: uniform continuity ¶
- depends_on proposition 7.22 Sequential characterization ¶ ↺
- depends_on theorem 7.7 Bolzano–Weierstrass ¶ ↺
- proves proof ch:05-real-analysis@proof-10 ¶
depends_on theorem 7.23 Intermediate value theorem ¶
- depends_on axiom 7.1 Completeness of $\R$ ¶ ↺
- depends_on definition 7.20 Continuity at a point ¶ ↺
- proves proof ch:05-real-analysis@proof-8 ¶
proves proof ch:04-topology@proof-6 ¶

Supports

depends_on definition 6.20 Winding number ¶
- depends_on example 6.21 ex:top-winding-values ¶
- depends_on lemma 6.22 Nearby loops wind alike ¶
  - depends_on proposition 6.23 The punctured plane is not simply connected ¶
depends_on lemma 6.22 Nearby loops wind alike ¶ ↺

Neighborhood

Every logical edge within two steps of this node.

declared and complete
partly declared
a check failed
not graded
declared in the source
inferred from structure

Edges

type	direction	node	provenance	where
`depends_on`	→	Extreme value theorem	declared	parts/02-mathematical-methods/04-topology.tex:318
`depends_on`	→	Heine–Cantor: uniform continuity	declared	parts/02-mathematical-methods/04-topology.tex:318
`depends_on`	→	Intermediate value theorem	declared	parts/02-mathematical-methods/04-topology.tex:318
`depends_on`	←	Winding number	declared	parts/02-mathematical-methods/04-topology.tex:381
`depends_on`	←	Nearby loops wind alike	declared	parts/02-mathematical-methods/04-topology.tex:400
`proves`	←	ch:04-topology@proof-6	declared	parts/02-mathematical-methods/04-topology.tex:321