theorem 7.23 Intermediate value theorem

open in the book · parts/02-mathematical-methods/05-real-analysis.tex:371 · p. 207

Rests on

depends_on axiom 7.1 Completeness of $\R$ ¶
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 definition 3.43 Map ¶
      - depends_on definition 3.24 Quantifiers ¶
      - depends_on definition 3.28 Set ¶
- depends_on equation 7.7 eq:ana-limit-left ¶
- depends_on equation 7.5 eq:ana-limit-right ¶
proves proof ch:05-real-analysis@proof-8 ¶

Supports

depends_on definition 7.74 $\pi$ ¶
- depends_on lemma 7.75 The first quadrant ¶
  - depends_on proposition 7.76 $\pi$ as the circle constant ¶
  - depends_on proposition 7.77 Special values, periodicity, and the kernel ¶
    - depends_on lemma 7.78 Chord, arc, tangent ¶
      - depends_on proposition 7.79 The polygon recursion ¶
    - depends_on proposition 7.85 Irrationality of $\pi$ ¶
      - depends_on remark 7.94 Three numbers, two kinds ¶
      - depends_on remark 7.87 Priority, and what irrationality does not give ¶
    - depends_on proposition 7.79 The polygon recursion ¶ ↺
    - depends_on proposition 7.83 Viète's product ¶
      - depends_on remark 7.84 One construction, two faces ¶
    - depends_on proposition 125.2 Crystallographic restriction ¶
      - depends_on remark 125.3 What the theorem does and does not forbid ¶
    - depends_on remark 7.88 $\pi$ elsewhere in this treatise ¶
  - depends_on proposition 125.2 Crystallographic restriction ¶ ↺
- depends_on proposition 7.76 $\pi$ as the circle constant ¶ ↺
- depends_on proposition 7.79 The polygon recursion ¶ ↺
- depends_on proposition 7.77 Special values, periodicity, and the kernel ¶ ↺
- depends_on proposition 125.2 Crystallographic restriction ¶ ↺
depends_on lemma 7.75 The first quadrant ¶ ↺
depends_on lemma 6.19 Continuous argument along a path ¶
- 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 ¶ ↺
depends_on proposition 7.32 Derivative of the inverse function ¶
- depends_on definition 9.140 Amplitude and the Jacobi elliptic functions ¶
  - depends_on proposition 9.143 A cubic quadrature between two turning points ¶
  - depends_on proposition 9.141 First properties ¶
    - depends_on proposition 9.143 A cubic quadrature between two turning points ¶ ↺
    - depends_on proposition 9.142 The differential equation of the sine amplitude ¶
  - depends_on proposition 9.144 The pendulum equation at finite amplitude ¶
- depends_on proposition 7.61 The logarithm ¶
  - depends_on corollary 7.69 $\ee$ is the unique self-reproducing base ¶
    - depends_on remark 7.70 What the condition says ¶
  - depends_on definition 7.62 Real powers ¶
    - depends_on definition 7.90 The golden ratio ¶
      - depends_on proposition 7.91 The golden ratio ¶
    - depends_on proposition 7.63 Laws of real powers ¶
      - depends_on definition 7.90 The golden ratio ¶ ↺
      - depends_on lemma 7.67 The limit over real exponents ¶
      - depends_on proposition 7.68 The natural base ¶
      - depends_on proposition 7.91 The golden ratio ¶ ↺
      - depends_on proposition 7.89 Irrationality of square roots ¶
    - depends_on proposition 7.89 Irrationality of square roots ¶ ↺
  - depends_on lemma 7.67 The limit over real exponents ¶ ↺
  - depends_on proposition 7.68 The natural base ¶ ↺
  - depends_on proposition 7.63 Laws of real powers ¶ ↺
- depends_on proposition 9.141 First properties ¶ ↺
- depends_on proposition 9.19 Separation of variables ¶
depends_on proposition 7.61 The logarithm ¶ ↺
depends_on proposition 7.76 $\pi$ as the circle constant ¶ ↺
depends_on theorem A.443 Sturm separation theorem ¶
depends_on theorem 11.101 Davies' bound ¶
- depends_on corollary 11.105 Trials factor at high significance ¶
- depends_on example 11.107 Degrading a local five sigma ¶
depends_on theorem 29.37 Reduction to one degree of freedom ¶
- depends_on phenomenon 29.38 Nutation of a released top ¶
- depends_on proposition 29.40 The sleeping top ¶
- depends_on remark 29.42 The closed form, and the mathematics it needs ¶

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`	→	Completeness of $\R$	declared	parts/02-mathematical-methods/05-real-analysis.tex:374
`depends_on`	→	Continuity at a point	declared	parts/02-mathematical-methods/05-real-analysis.tex:374
`depends_on`	←	$\pi$	declared	parts/02-mathematical-methods/05-real-analysis.tex:1947
`depends_on`	←	The first quadrant	declared	parts/02-mathematical-methods/05-real-analysis.tex:1976
`depends_on`	←	Continuous argument along a path	declared	parts/02-mathematical-methods/04-topology.tex:318
`depends_on`	←	Derivative of the inverse function	declared	parts/02-mathematical-methods/05-real-analysis.tex:625
`depends_on`	←	The logarithm	declared	parts/02-mathematical-methods/05-real-analysis.tex:1397
`depends_on`	←	$\pi$ as the circle constant	declared	parts/02-mathematical-methods/05-real-analysis.tex:2007
`depends_on`	←	Sturm separation theorem	declared	appendices/A-long-proofs.tex:21785
`depends_on`	←	Davies' bound	declared	parts/02-mathematical-methods/09-probability-statistics.tex:3101
`depends_on`	←	Reduction to one degree of freedom	declared	parts/03-classical-mechanics/12-rigid-body-rotating-frames.tex:1321
`proves`	←	ch:05-real-analysis@proof-8	declared	parts/02-mathematical-methods/05-real-analysis.tex:377