theorem 7.40 Continuous functions are integrable

open in the book · parts/02-mathematical-methods/05-real-analysis.tex:785 · p. 216

Rests on

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 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 ¶
proves proof ch:05-real-analysis@proof-23 ¶

Supports

depends_on corollary A.502 All iterated orders agree ¶
- depends_on lemma A.507 Coordinate permutations ¶
depends_on definition 16.11 Admissible class; functional ¶
- depends_on definition 16.59 Weierstrass excess function ¶
- depends_on definition 16.12 The two norms; weak and strong neighbourhoods ¶
  - depends_on definition 16.13 Weak and strong extrema ¶
    - depends_on example 16.14 A weak minimum that is not strong ¶
    - depends_on proposition 16.16 Stationarity is necessary ¶
      - depends_on theorem 16.22 Euler–Lagrange ¶
      - depends_on theorem 16.51 Legendre's necessary condition ¶
    - depends_on theorem 16.60 Weierstrass' necessary condition ¶
- depends_on definition 16.15 Variation; the first variation ¶
  - depends_on definition 16.83 One-parameter transformation group; variational symmetry ¶
  - depends_on lemma 16.18 Fundamental lemma ¶
    - depends_on theorem A.688 The Kirchhoff plate equation ¶
      - depends_on remark A.690 Poisson's count, and why it cannot be right ¶
      - depends_on theorem A.691 Kirchhoff's edge conditions and the corner force ¶
    - depends_on theorem 16.36 Euler–Lagrange equations for several independent variables ¶
      - depends_on definition 44.6 Stress–energy tensor ¶
      - depends_on proposition 28.42 The wave equation of a stretched string ¶
    - depends_on theorem 16.38 Euler–Poisson equation ¶
      - depends_on theorem 30.56 The beam equation ¶
      - depends_on theorem 30.64 The Kirchhoff plate equation ¶
    - depends_on theorem 16.41 Multiplier rule for pointwise constraints ¶
      - depends_on proposition 21.56 The two prescriptions and their difference ¶
    - depends_on theorem 16.31 Natural boundary condition ¶
      - depends_on remark 30.67 Germain, Kirchhoff, and the edge conditions ¶
      - depends_on theorem A.691 Kirchhoff's edge conditions and the corner force ¶ ↺
      - depends_on theorem 16.32 Transversality ¶
    - depends_on theorem 25.19 Hamilton's equations from the first-order action ¶
    - depends_on theorem 25.22 Faddeev–Jackiw equations and brackets ¶
      - depends_on example 25.24 A charged particle in a strong magnetic field ¶
  - depends_on proposition 16.16 Stationarity is necessary ¶ ↺
  - 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 proposition 16.68 Weierstrass' counterexample ¶
depends_on definition 9.137 Elliptic integrals of the three kinds ¶
- 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 9.139 Series for the complete integral of the first kind ¶
  - depends_on proposition 9.144 The pendulum equation at finite amplitude ¶ ↺
depends_on lemma A.501 Iterated integration over a box ¶
- depends_on corollary A.502 All iterated orders agree ¶ ↺
- depends_on lemma A.509 Primitive maps have the substitution property ¶
  - depends_on proposition A.511 The substitution property is universal ¶
    - depends_on remark A.516 The hypotheses of the global form ¶
    - depends_on theorem A.513 Change of variables ¶
      - depends_on corollary A.514 Degeneracy on a negligible set ¶
      - depends_on remark A.516 The hypotheses of the global form ¶ ↺
depends_on lemma 16.18 Fundamental lemma ¶ ↺
depends_on remark 7.128 What the derivations below take as given ¶
- depends_on lemma A.312 The computation in one chart ¶
- depends_on theorem 7.129 Change of variables in a multiple integral ¶
  - depends_on example 7.130 The two Jacobians this treatise uses ¶
    - depends_on lemma A.518 The excised ball ¶
      - depends_on proposition A.521 Every derivative passes onto $g$ ¶
      - depends_on proposition A.519 The convolution exists ¶
  - depends_on lemma A.518 The excised ball ¶ ↺
  - depends_on proposition A.519 The convolution exists ¶ ↺
- depends_on theorem 7.133 Gauss ¶
  - depends_on lemma A.708 The disturbance flux vanishes ¶
    - depends_on corollary A.711 The disturbance is a dipole at leading order ¶
  - depends_on lemma A.712 The force is a far-field integral ¶
  - depends_on lemma A.117 Laplacian in orthogonal coordinates ¶
    - depends_on theorem A.120 Stäckel and Robertson conditions ¶
      - depends_on example A.122 Cartesian ¶
      - depends_on example A.123 Circular cylindrical ¶
      - depends_on example A.124 Spherical ¶
      - depends_on proposition A.128 The ellipsoidal system separates ¶
  - 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 31.11 Transport theorem for a material volume ¶
    - depends_on theorem 31.12 Conservation of mass ¶
      - depends_on corollary 31.13 Incompressible flow ¶
      - depends_on phenomenon 31.88 Sound travels at the adiabatic speed ¶
      - depends_on proposition 52.24 The p-mode cavity ¶
      - depends_on theorem 31.91 Rankine–Hugoniot jump conditions ¶
    - depends_on theorem 31.22 Euler's equations of motion ¶
      - depends_on lemma A.712 The force is a far-field integral ¶ ↺
      - depends_on phenomenon 31.88 Sound travels at the adiabatic speed ¶ ↺
      - depends_on proposition A.797 The outer limit loses a boundary condition ¶
      - depends_on proposition 31.80 Rayleigh's circulation criterion ¶
      - depends_on proposition 52.24 The p-mode cavity ¶ ↺
      - depends_on remark 31.23 Boundary conditions, and what the ideal model omits ¶
      - depends_on theorem 31.24 Bernoulli ¶
      - depends_on theorem 31.64 Kelvin's circulation theorem ¶
      - depends_on theorem 31.37 The Navier–Stokes equations ¶
      - depends_on theorem 31.91 Rankine–Hugoniot jump conditions ¶ ↺
      - depends_on theorem 31.77 Rayleigh's inflexion-point criterion ¶
  - depends_on lemma 106.86 A shift of a divergent integral leaves a surface term ¶
    - depends_on proposition 106.88 The obstruction ¶
      - depends_on theorem 106.95 The cancellation condition ¶
  - depends_on lemma 10.44 Green's identities ¶
    - depends_on proposition 10.71 Green's representation formula ¶
    - depends_on proposition 10.67 The Newtonian potential is the fundamental solution ¶
      - depends_on example 10.49 The method of images ¶
      - depends_on lemma A.94 Fundamental solution of $\pp_{\bar z}$ ¶
      - depends_on proposition A.522 The Newtonian potential inverts the Laplacian ¶
      - depends_on proposition 10.71 Green's representation formula ¶ ↺
      - depends_on remark 10.73 What compact support is doing, and the rate that replaces it ¶
      - depends_on theorem 7.137 Helmholtz decomposition ¶
    - depends_on theorem 10.45 Symmetry of the Green's function ¶
  - depends_on lemma 10.58 Darboux's equation for spherical means ¶
    - depends_on theorem 10.59 Kirchhoff's formula ¶
      - depends_on corollary 10.60 Huygens' principle in three space dimensions ¶
    - depends_on theorem 10.69 Mean-value property ¶
      - depends_on corollary A.145 Harmonic functions are smooth ¶
      - depends_on theorem A.135 Converse of the mean-value property ¶
      - depends_on theorem 10.77 Strong maximum principle for harmonic functions ¶
  - depends_on proposition 23.54 The geometrical amplitude diverges ¶
    - depends_on proposition 23.55 The fold, and the quarter-power law ¶
      - depends_on proposition 23.56 The phase jump at a fold ¶
      - depends_on remark 23.58 The caustics that are not turning surfaces ¶
    - depends_on remark 23.59 What is observed ¶
  - depends_on proposition 10.57 Energy in a backward cone ¶
    - depends_on proposition 28.63 Energy density, flux, and the conservation law ¶
  - depends_on theorem 16.36 Euler–Lagrange equations for several independent variables ¶ ↺
  - depends_on theorem 32.6 Evolution of phase volume ¶
    - depends_on definition 32.7 Conservative and dissipative flows ¶
      - depends_on definition 32.62 The Lorenz system ¶
    - depends_on definition 32.62 The Lorenz system ¶ ↺
    - depends_on proposition 32.64 The Lorenz flow contracts volume ¶
      - depends_on phenomenon 32.68 Strange attractors ¶
    - depends_on proposition 32.76 The exponents sum to the mean divergence ¶
  - … 1 more
- depends_on theorem 7.131 Green ¶
  - depends_on proposition 32.33 Bendixson's negative criterion ¶
  - depends_on proposition 9.36 Bendixson–Dulac negative criterion ¶
  - depends_on theorem 7.132 Stokes ¶
    - depends_on definition 31.63 Circulation ¶
      - depends_on theorem 31.64 Kelvin's circulation theorem ¶ ↺
    - depends_on proposition 7.136 Properties of conservative fields ¶
      - depends_on example 31.15 The stream function of a plane flow ¶
      - depends_on proposition 31.28 Potential flow reduces to Laplace's equation ¶
      - depends_on theorem 16.63 Hilbert's invariant integral ¶
      - depends_on theorem 30.15 Compatibility is sufficient on a simply connected body ¶
  - depends_on theorem 8.12 Cauchy ¶
    - depends_on corollary 8.15 Deformation of contours ¶
      - depends_on lemma 106.1 The short-time kernel ¶
      - depends_on theorem 8.16 Cauchy integral formula ¶
      - depends_on theorem 8.21 Laurent expansion ¶
      - depends_on theorem 8.24 Residue theorem ¶
    - depends_on corollary 17.71 Residue evaluation and causality ¶
      - depends_on corollary 17.76 Heaviside's expansion theorem ¶
      - depends_on proposition 17.77 Poles and stability ¶
    - depends_on lemma 100.27 Wick rotation ¶
      - depends_on lemma 100.28 The $d$-dimensional loop integral ¶
    - depends_on proposition 106.14 The Feynman contour permits the rotation ¶
    - 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 17.94 Mellin inversion ¶
      - depends_on proposition 17.96 Poles give asymptotics ¶
      - depends_on theorem 17.100 Moments diagonalise a convolution evolution ¶
  - depends_on theorem 13.41 Holonomy equals the enclosed curvature; local Gauss–Bonnet ¶
    - depends_on example 13.42 The sphere, the solid angle, and the pole ¶
  - depends_on theorem 10.96 Rankine–Hugoniot condition ¶
    - depends_on definition 10.99 Entropy condition ¶
      - depends_on definition 10.101 Entropy pair; entropy inequality ¶
    - depends_on example 10.97 Burgers shock ¶
    - depends_on example 10.98 Weak solutions are not unique ¶
    - depends_on proposition 10.102 Jump form of the entropy inequality ¶
- depends_on theorem A.303 Change of variables for multiple integrals; quoted ¶
  - depends_on lemma A.305 The chart integral is well defined ¶
    - depends_on definition A.308 Integral over the manifold ¶
      - depends_on theorem A.311 General Stokes theorem ¶
depends_on theorem 7.42 Fundamental theorem of calculus, I ¶
- depends_on corollary 7.110 Variable limits of integration ¶
- depends_on definition A.450 The space $W^{1,r}(a,b)$ ¶
  - depends_on definition A.462 The weighted space and the energy space ¶
    - 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 theorem A.456 Convexity in $q$ implies weak lower semicontinuity ¶
  - 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.453 Uniform bound and uniform Hölder continuity ¶ ↺
- depends_on lemma A.520 A $C^{1}$ limit ¶
  - depends_on proposition A.521 Every derivative passes onto $g$ ¶ ↺
- depends_on lemma A.440 Grönwall's inequality ¶
  - depends_on theorem A.441 Continuous dependence on the data and on the initial point ¶
    - depends_on corollary A.445 Displacing the initial point ¶
- depends_on lemma 16.19 du Bois-Reymond ¶
  - 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 theorem 16.43 Euler's rule for integral constraints ¶ ↺
- depends_on lemma 9.10 Grönwall's inequality ¶
  - depends_on proposition 9.11 Continuous dependence on the initial data ¶
  - depends_on proposition 9.31 Lyapunov's first method: asymptotic stability ¶
- depends_on proposition 11.22 Properties of the distribution function ¶
  - depends_on proposition 11.27 Probability integral transform ¶
- depends_on proposition 9.18 The linear equation: integrating factor ¶
  - 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 9.19 Separation of variables ¶
- depends_on theorem 7.43 Fundamental theorem of calculus, II ¶
  - 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 lemma 9.136 Wallis integrals ¶
      - depends_on proposition 9.139 Series for the complete integral of the first kind ¶ ↺
    - 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 corollary 16.23 du Bois-Reymond form ¶
    - depends_on corollary 16.33 Weierstrass–Erdmann corner conditions ¶
      - depends_on theorem 16.55 Jacobi's necessary condition ¶
  - depends_on lemma 7.116 Functions vanishing on a regular zero set ¶
  - depends_on lemma A.72 Iterated integral inequality ¶
    - depends_on theorem A.74 Flow of a time-dependent vector field ¶
      - depends_on theorem 13.125 Existence, uniqueness and smoothness of the flow ¶
  - depends_on lemma A.520 A $C^{1}$ limit ¶ ↺
  - depends_on lemma A.172 Dirichlet ¶
    - depends_on lemma A.193 A cosine integral ¶
      - depends_on lemma A.194 The Fejér kernel and its transform ¶
    - depends_on theorem A.177 Fourier inversion for a distribution function ¶
      - depends_on corollary A.178 The characteristic function determines the distribution ¶
      - depends_on lemma A.196 Inversion for a difference ¶
  - depends_on lemma A.221 Counting identity ¶
  - depends_on lemma A.312 The computation in one chart ¶ ↺
  - 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 8.10 Fundamental theorem for contours ¶
    - depends_on theorem 8.13 Goursat ¶
    - depends_on theorem 8.18 Liouville ¶
      - depends_on corollary A.723 Potential of one infinite row ¶
      - depends_on lemma A.608 An entire function with a quadratic bound on its real part ¶
      - depends_on theorem 8.19 Fundamental theorem of algebra ¶
      - depends_on theorem 12.54 The spectrum is compact and non-empty ¶
  - depends_on proposition 17.41 The identities physics uses ¶
    - depends_on proposition 17.90 Hankel transform as the axially symmetric Fourier transform ¶
    - depends_on proposition 17.52 Linear translation-invariant systems ¶
      - depends_on definition 28.25 Complex susceptibility ¶
      - depends_on proposition 17.61 Mean-square response of a damped resonator ¶
  - depends_on proposition 17.73 Initial- and final-value theorems ¶
  - … 4 more
- depends_on theorem 10.52 Duhamel's principle ¶
  - depends_on corollary 10.53 Duhamel for the wave equation ¶

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Edges

type	direction	node	provenance	where
`depends_on`	→	Darboux sums and the definite integral	declared	parts/02-mathematical-methods/05-real-analysis.tex:788
`depends_on`	→	Extreme value theorem	declared	parts/02-mathematical-methods/05-real-analysis.tex:788
`depends_on`	→	Heine–Cantor: uniform continuity	declared	parts/02-mathematical-methods/05-real-analysis.tex:788
`depends_on`	←	All iterated orders agree	declared	appendices/A-long-proofs.tex:24637
`depends_on`	←	Admissible class; functional	declared	parts/02-mathematical-methods/14-calculus-of-variations.tex:386
`depends_on`	←	Elliptic integrals of the three kinds	declared	parts/02-mathematical-methods/07-odes-sturm-liouville.tex:5250
`depends_on`	←	Iterated integration over a box	declared	appendices/A-long-proofs.tex:24598
`depends_on`	←	Fundamental lemma	declared	parts/02-mathematical-methods/14-calculus-of-variations.tex:544
`depends_on`	←	What the derivations below take as given	declared	parts/02-mathematical-methods/05-real-analysis.tex:4261
`depends_on`	←	Fundamental theorem of calculus, I	declared	parts/02-mathematical-methods/05-real-analysis.tex:834
`proves`	←	ch:05-real-analysis@proof-23	declared	parts/02-mathematical-methods/05-real-analysis.tex:791