proposition 9.11 Continuous dependence on the initial data

Rests on

• depends_on equation 9.4 eq:slt-lipschitz ¶
• depends_on lemma 9.10 Grönwall's inequality ¶
  • depends_on corollary 7.36 cor:ana-mvt-consequences ¶
    • 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-20 ¶
  • 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 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 ¶
  • proves proof ch:07-odes-sturm-liouville@proof-4 ¶
• 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 definition 7.16 Limit ¶ ↺
      • depends_on equation 7.7 eq:ana-limit-left ¶
      • depends_on equation 7.5 eq:ana-limit-right ¶
    • depends_on proposition 7.47 Comparison; absolute convergence ¶
      • 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 ¶
      • depends_on theorem 7.8 Cauchy criterion ¶
        • depends_on corollary A.47 Cauchy completeness ¶
        • depends_on definition 7.4 Convergence ¶ ↺
        • proves proof ch:05-real-analysis@proof-5 ¶
      • proves proof ch:05-real-analysis@proof-28 ¶
    • 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-5 ¶

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