theorem A.441 Continuous dependence on the data and on the initial point

open in the book · appendices/A-long-proofs.tex:21635 · p. 3008

Rests on

depends_on equation 9.7 eq:slt-picard-integral ¶
depends_on lemma A.440 Grönwall's inequality ¶
- depends_on theorem 7.42 Fundamental theorem of calculus, I ¶
  - depends_on definition 7.26 Derivative of a function 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 7.9 Real function ¶ ↺
  - depends_on theorem 7.40 Continuous functions are integrable ¶
    - depends_on definition 7.39 Darboux sums and the definite integral ¶
      - depends_on axiom 7.1 Completeness of $\R$ ¶
    - 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 theorem 7.7 Bolzano–Weierstrass ¶
      - 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 ¶
    - proves proof ch:05-real-analysis@proof-23 ¶
  - depends_on theorem 7.24 Extreme value theorem ¶ ↺
  - proves proof ch:05-real-analysis@proof-24 ¶
- 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 proposition 7.3 Triangle inequality ¶
      - 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 ¶
      - proves proof ch:05-real-analysis@proof-17 ¶
    - 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 app:A-long-proofs@proof-262 ¶
depends_on lemma A.439 Existence, uniqueness, and the structure of the zeros ¶
- depends_on corollary 9.9 Linear equations: existence on the whole interval ¶
  - depends_on definition 9.2 Linear equation; homogeneity ¶
    - depends_on definition 9.1 Ordinary differential equation ¶
  - depends_on remark 9.3 Normal form and first-order systems ¶
  - depends_on theorem 9.8 Picard–Lindelöf ¶
    - depends_on definition 9.4 Lipschitz condition ¶
    - depends_on equation 9.3 eq:slt-ode-system ¶
    - depends_on lemma 9.6 Weierstrass $M$-test; uniform limits are continuous ¶
      - depends_on definition 7.20 Continuity at a point ¶
      - depends_on proposition 7.47 Comparison; absolute convergence ¶
      - proves proof ch:07-odes-sturm-liouville@proof-1 ¶
    - proves proof ch:07-odes-sturm-liouville@proof-2 ¶
  - proves proof ch:07-odes-sturm-liouville@proof-3 ¶
- depends_on definition A.438 Momentum variable and the accessory system ¶
  - depends_on equation 16.49 eq:calcvar-jacobi ¶
  - depends_on equation 9.3 eq:slt-ode-system ¶ ↺
- proves proof app:A-long-proofs@proof-261 ¶
proves proof app:A-long-proofs@proof-263 ¶

Supports

depends_on corollary A.445 Displacing the initial point ¶

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`	→	eq:slt-picard-integral	declared	appendices/A-long-proofs.tex:21656
`depends_on`	→	Grönwall's inequality	declared	appendices/A-long-proofs.tex:21656
`depends_on`	→	Existence, uniqueness, and the structure of the zeros	declared	appendices/A-long-proofs.tex:21656
`depends_on`	←	Displacing the initial point	declared	appendices/A-long-proofs.tex:21968
`proves`	←	app:A-long-proofs@proof-263	declared	appendices/A-long-proofs.tex:21660