proposition 7.27 Differentiable implies continuous

open in the book · parts/02-mathematical-methods/05-real-analysis.tex:455 · p. 208

Rests on

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 ¶
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 proposition 7.6 Algebra of limits ¶
- depends_on definition 7.4 Convergence ¶
  - depends_on definition 7.2 Absolute value ¶ ↺
- depends_on proposition 7.3 Triangle inequality ¶
  - depends_on definition 7.2 Absolute value ¶ ↺
  - proves proof ch:05-real-analysis@proof-1 ¶
- proves proof ch:05-real-analysis@proof-3 ¶
proves proof ch:05-real-analysis@proof-11 ¶

Supports

depends_on proposition 7.31 Chain rule ¶
- depends_on corollary 7.44 Substitution and integration by parts ¶
  - depends_on definition 8.8 Contour integral ¶
  - depends_on lemma 16.20 Mixed form ¶
    - depends_on theorem 16.35 System of Euler–Lagrange equations ¶
      - depends_on proposition 16.6 The isoperimetric extremal is a circle ¶
      - depends_on theorem 16.41 Multiplier rule for pointwise constraints ¶
    - depends_on theorem 16.22 Euler–Lagrange ¶
      - depends_on definition 16.62 Field of extremals; slope function ¶
      - depends_on phenomenon 30.52 Euler buckling ¶
      - depends_on proposition 29.39 Steady precession ¶
      - depends_on theorem 16.35 System of Euler–Lagrange equations ¶ ↺
      - depends_on theorem 16.31 Natural boundary condition ¶
    - depends_on theorem 16.43 Euler's rule for integral constraints ¶
      - depends_on example 16.45 Two solved isoperimetric problems ¶
      - depends_on proposition 16.4 The hanging chain is a catenary ¶
      - depends_on proposition 16.6 The isoperimetric extremal is a circle ¶ ↺
  - depends_on lemma 9.136 Wallis integrals ¶
    - depends_on proposition 9.139 Series for the complete integral of the first kind ¶
      - depends_on proposition 9.144 The pendulum equation at finite amplitude ¶
  - depends_on proposition 7.145 Delta as a limit; elementary properties ¶
  - depends_on proposition 16.50 The second variation ¶
  - depends_on proposition 5.95 The focal polar equation of a conic ¶
  - depends_on proposition 9.143 A cubic quadrature between two turning points ¶
  - depends_on proposition 9.144 The pendulum equation at finite amplitude ¶ ↺
  - depends_on theorem 16.38 Euler–Poisson equation ¶
    - depends_on theorem 30.56 The beam equation ¶
      - depends_on example 30.58 A steel ruler ¶
      - depends_on proposition 30.57 Bending waves are dispersive ¶
    - depends_on theorem 30.64 The Kirchhoff plate equation ¶
      - depends_on phenomenon 30.65 Chladni figures ¶
      - depends_on proposition 30.66 Chladni scaling ¶
      - depends_on remark 30.67 Germain, Kirchhoff, and the edge conditions ¶
- depends_on lemma A.479 Conformal invariance of the Dirichlet integral ¶
  - depends_on lemma A.487 The normalisation costs nothing ¶
    - depends_on theorem A.492 Douglas' conformality theorem; quoted ¶
  - depends_on lemma A.485 Conformal automorphisms of the disc ¶
    - depends_on definition A.486 The normalised admissible class ¶
    - depends_on lemma A.487 The normalisation costs nothing ¶ ↺
- depends_on lemma 27.15 Velocity and acceleration in plane polar coordinates ¶
- depends_on proposition 7.111 Jacobi's formula, cofactor form ¶
- depends_on proposition 7.63 Laws of real powers ¶
  - depends_on definition 7.90 The golden ratio ¶
    - depends_on proposition 7.91 The golden ratio ¶
      - depends_on proposition 7.92 Continued fraction and the Fibonacci ratios ¶
      - depends_on proposition 125.2 Crystallographic restriction ¶
      - depends_on remark 7.95 Where $\varphi$ is used ¶
      - depends_on remark 7.94 Three numbers, two kinds ¶
  - depends_on lemma 7.67 The limit over real exponents ¶
    - depends_on proposition 7.68 The natural base ¶
      - depends_on remark 7.70 What the condition says ¶
  - 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 7.91 The golden ratio ¶ ↺
- depends_on proposition 16.26 Covariance under change of dependent variable ¶
- depends_on proposition 10.15 Method of characteristics, first order ¶
  - depends_on definition 10.93 Scalar conservation law ¶
    - depends_on definition A.150 Kruzhkov entropy pair ¶
      - depends_on lemma A.157 Two elementary identities ¶
      - depends_on proposition A.156 The Kruzhkov inequality ¶
      - depends_on theorem A.151 $L^{1}$ contraction on a cone ¶
    - depends_on definition 10.95 Weak solution of a conservation law ¶
      - depends_on definition 10.99 Entropy condition ¶
      - depends_on definition 10.101 Entropy pair; entropy inequality ¶
      - depends_on example 10.98 Weak solutions are not unique ¶
      - depends_on theorem A.151 $L^{1}$ contraction on a cone ¶ ↺
      - depends_on theorem 10.103 Uniqueness in the entropy class ¶
      - depends_on theorem 10.96 Rankine–Hugoniot condition ¶
    - depends_on example 10.97 Burgers shock ¶
    - depends_on proposition 10.94 Gradient catastrophe ¶
  - depends_on proposition 10.16 Cauchy's characteristic strips ¶
  - depends_on proposition 10.94 Gradient catastrophe ¶ ↺
  - depends_on remark 31.90 Why a compression wave must steepen ¶
  - depends_on remark 23.3 What Part II owes this chapter ¶
- depends_on theorem 16.29 Beltrami identity ¶
  - depends_on proposition 16.105 The optical invariant in a stratified medium ¶
  - depends_on proposition 16.39 Ostrogradsky instability ¶
- depends_on theorem 10.8 The type is a coordinate invariant ¶
  - depends_on lemma A.90 Affine normalization ¶
    - depends_on lemma A.91 Smallness after rescaling ¶
  - depends_on proposition 10.13 Canonical form in two variables ¶
    - depends_on theorem A.83 Local canonical form for an elliptic operator with Hölder coefficients ¶
    - depends_on theorem 10.55 d'Alembert's formula ¶
      - depends_on corollary 10.56 Domain of dependence and influence ¶
      - depends_on theorem 10.59 Kirchhoff's formula ¶
  - depends_on proposition 10.11 What a characteristic surface is ¶
    - depends_on corollary 10.12 Characteristics of the three canonical operators ¶
      - depends_on corollary 10.60 Huygens' principle in three space dimensions ¶
    - depends_on definition A.102 Normal Cauchy problem of order $k$ ¶
      - depends_on definition A.103 First-order quasilinear system with zero data ¶
      - depends_on lemma A.105 Formal reduction of a normal problem ¶
    - depends_on theorem 10.22 Cauchy–Kovalevskaya ¶
depends_on proposition 7.68 The natural base ¶ ↺
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 7.63 Laws of real powers ¶ ↺
  - 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 7.30 Leibniz rule ¶
- depends_on corollary 7.44 Substitution and integration by parts ¶ ↺
- depends_on proposition 7.108 Euler's theorem on homogeneous functions ¶
- depends_on proposition 7.111 Jacobi's formula, cofactor form ¶ ↺
- depends_on proposition 7.122 Product identities of the nabla calculus ¶
- depends_on proposition 5.2 Jacobi's formula, column form ¶
  - depends_on lemma 44.10 Divergence theorem on $(M,g)$ ¶
    - depends_on proposition 44.11 The boundary term, and the Gibbons–Hawking–York action ¶
  - depends_on lemma 44.7 Variation of inverse and determinant ¶
    - depends_on proposition 44.17 Stress–energy of the electromagnetic field ¶
      - depends_on proposition 44.38 The electromagnetic field satisfies NEC, WEC, SEC and DEC ¶
    - depends_on theorem 44.9 Variation of the Einstein–Hilbert action ¶
      - depends_on proposition 44.11 The boundary term, and the Gibbons–Hawking–York action ¶ ↺
      - depends_on proposition 44.30 The cosmological term ¶
  - depends_on proposition 7.111 Jacobi's formula, cofactor form ¶ ↺
- depends_on proposition 9.28 The scalar constant-coefficient equation ¶
  - depends_on proposition 9.88 The model problem, and the error of the composite ¶
- depends_on theorem 7.38 Taylor's theorem with Lagrange remainder ¶
  - depends_on definition 32.57 The Hénon–Heiles Hamiltonian ¶
  - depends_on definition 32.12 Linearization ¶
    - depends_on proposition 32.17 Classification of planar fixed points ¶
      - depends_on example 32.18 A linear centre that is really a stable focus ¶
      - depends_on theorem 32.25 Hopf bifurcation, quoted ¶
    - depends_on theorem 32.15 Hartman–Grobman, restated from Part II ¶
      - depends_on definition 32.22 Bifurcation ¶
      - depends_on example 32.18 A linear centre that is really a stable focus ¶ ↺
    - depends_on theorem 32.13 Linear stability ¶
      - depends_on proposition 32.65 Fixed points of the Lorenz system and their stability ¶
      - depends_on proposition 32.23 Saddle-node bifurcation ¶
      - depends_on proposition 32.24 Transcritical and pitchfork bifurcations ¶
      - depends_on proposition 28.69 Which branch is stable, and where it ends ¶
  - depends_on lemma A.754 Exact depth expansion ¶
    - depends_on proposition A.755 The Boussinesq pair ¶
      - depends_on theorem A.756 The Korteweg–de Vries equation ¶
  - depends_on lemma A.187 Exponentiation ¶
  - depends_on lemma A.138 The flat exponential ¶
    - depends_on definition A.139 The standard radial mollifier ¶
      - depends_on definition A.141 Mollification ¶
      - depends_on lemma A.140 Properties of the mollifier ¶
    - depends_on lemma A.140 Properties of the mollifier ¶ ↺
  - depends_on lemma A.452 Young and Hölder ¶
    - depends_on lemma A.489 Courant–Lebesgue ¶
      - depends_on theorem A.490 Equicontinuity of the normalised class ¶
    - depends_on lemma A.463 Poincaré inequality; $B_{K}$ is an inner product ¶
      - depends_on lemma A.464 $H_{E}$ is a Hilbert space ¶
      - depends_on lemma A.466 The homogeneous Dirichlet problem is trivial ¶
      - depends_on theorem A.469 The Green operator inverts $L_{K}$ ¶
    - depends_on lemma A.453 Uniform bound and uniform Hölder continuity ¶
      - depends_on theorem A.455 Compact embedding of $W^{1,r}(a,b)$ into the continuous functions ¶
  - depends_on lemma A.212 Second-order flatness ¶
    - depends_on lemma A.213 The three minima ¶
      - depends_on theorem A.214 Wilks, $k$ parameters and $r$ constraints ¶
  - depends_on lemma A.209 Uniform third-order remainder ¶
    - depends_on corollary A.210 Uniform quadratic approximation on the $n^{-1/2}$ scale ¶
      - depends_on lemma A.213 The three minima ¶ ↺
    - depends_on lemma A.211 Root-$n$ localisation ¶
      - depends_on lemma A.213 The three minima ¶ ↺
  - depends_on phenomenon 28.13 Universality of small oscillations ¶
  - depends_on proposition 16.50 The second variation ¶ ↺
  - depends_on proposition 17.67 Holomorphy ¶
    - depends_on theorem 17.78 Causality implies dispersion relations ¶
      - depends_on remark 33.10 Fibre anelasticity, and why it biases one method and not the other ¶
      - depends_on remark 28.26 Causality is visible in the susceptibility ¶
      - depends_on remark 28.62 Neither velocity is a speed limit ¶
    - depends_on theorem 17.93 Fundamental strip and holomorphy ¶
      - depends_on theorem 17.94 Mellin inversion ¶
  - depends_on proposition 9.88 The model problem, and the error of the composite ¶ ↺
  - … 2 more
depends_on theorem 7.51 Termwise differentiation ¶
- 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 7.85 Irrationality of $\pi$ ¶
      - depends_on proposition 7.79 The polygon recursion ¶
      - depends_on proposition 7.83 Viète's product ¶
      - depends_on proposition 125.2 Crystallographic restriction ¶ ↺
      - 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 7.93 A polynomial has at most $n$ roots ¶
- depends_on lemma 7.71 Derivatives; the Pythagorean identity ¶
  - depends_on lemma 7.75 The first quadrant ¶ ↺
  - depends_on lemma 7.78 Chord, arc, tangent ¶ ↺
  - depends_on lemma 9.136 Wallis integrals ¶ ↺
  - depends_on proposition 7.76 $\pi$ as the circle constant ¶ ↺
  - depends_on proposition 7.85 Irrationality of $\pi$ ¶ ↺
  - depends_on proposition 125.2 Crystallographic restriction ¶ ↺
  - 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 remark 7.80 Why these are the polygon perimeters ¶
- depends_on lemma 7.75 The first quadrant ¶ ↺
- depends_on proposition 9.22 The exponential and its derivative ¶
  - depends_on theorem 9.23 Solution of a constant-coefficient system ¶
    - depends_on corollary 9.25 Normal modes of a diagonalizable system ¶
      - depends_on corollary 9.27 Linear stability ¶
      - depends_on proposition 9.29 The planar classification ¶
    - depends_on proposition 9.87 The distinguished limit, and which end carries the layer ¶
      - depends_on proposition 9.88 The model problem, and the error of the composite ¶ ↺
    - depends_on proposition 9.31 Lyapunov's first method: asymptotic stability ¶
    - depends_on proposition 9.43 The unmodulated line, and the feet of the tongues ¶
    - depends_on theorem 9.32 Hartman–Grobman; quoted ¶
      - depends_on theorem 32.15 Hartman–Grobman, restated from Part II ¶ ↺
    - depends_on theorem 9.26 Structure of the solutions ¶
      - depends_on corollary 9.27 Linear stability ¶ ↺

Neighborhood

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Edges

type	direction	node	provenance	where
`depends_on`	→	Continuity at a point	declared	parts/02-mathematical-methods/05-real-analysis.tex:458
`depends_on`	→	Derivative of a function at a point	declared	parts/02-mathematical-methods/05-real-analysis.tex:458
`depends_on`	→	Algebra of limits	declared	parts/02-mathematical-methods/05-real-analysis.tex:458
`depends_on`	←	Chain rule	declared	parts/02-mathematical-methods/05-real-analysis.tex:589
`depends_on`	←	The natural base	declared	parts/02-mathematical-methods/05-real-analysis.tex:1628
`depends_on`	←	The logarithm	declared	parts/02-mathematical-methods/05-real-analysis.tex:1397
`depends_on`	←	Leibniz rule	declared	parts/02-mathematical-methods/05-real-analysis.tex:562
`depends_on`	←	Termwise differentiation	declared	parts/02-mathematical-methods/05-real-analysis.tex:1018
`proves`	←	ch:05-real-analysis@proof-11	declared	parts/02-mathematical-methods/05-real-analysis.tex:461