theorem 7.25 Heine–Cantor: uniform continuity

open in the book · parts/02-mathematical-methods/05-real-analysis.tex:409 · p. 208

Rests on

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 definition 3.43 Map ¶
  - 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 $\R$ ¶
      - depends_on definition A.36 Cut ¶ ↺
    - 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-10 ¶

Supports

depends_on lemma A.195 Riemann–Lebesgue, continuous compactly supported case ¶
- depends_on lemma A.196 Inversion for a difference ¶
  - depends_on theorem A.198 Esseen's smoothing inequality ¶
depends_on lemma A.500 Graphs and $C^{1}$ images have zero content ¶
- depends_on corollary A.514 Degeneracy on a negligible set ¶
- depends_on lemma A.512 The boundary strip is thin ¶
  - 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 A.499 What zero content buys ¶
- depends_on corollary A.514 Degeneracy on a negligible set ¶ ↺
- depends_on lemma A.500 Graphs and $C^{1}$ images have zero content ¶ ↺
- depends_on theorem A.513 Change of variables ¶ ↺
depends_on lemma A.512 The boundary strip is thin ¶ ↺
depends_on lemma A.73 Differentiation under the integral sign ¶
- depends_on lemma A.77 Poincaré lemma, converse form ¶
depends_on lemma A.176 Helly–Bray ¶
- depends_on corollary A.181 The converse ¶
depends_on lemma A.488 Small chords cut off small arcs ¶
- depends_on theorem A.490 Equicontinuity of the normalised class ¶
depends_on lemma 14.34 The $n$-sphere is simply connected for $n \ge 2$ ¶
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 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 definition 44.6 Stress–energy tensor ¶
      - depends_on definition 44.36 Energy conditions ¶
      - depends_on proposition 44.17 Stress–energy of the electromagnetic field ¶
      - depends_on theorem 44.19 Covariant conservation of stress–energy ¶
      - depends_on theorem 44.9 Variation of the Einstein–Hilbert action ¶
    - depends_on proposition 28.42 The wave equation of a stretched string ¶
  - 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 7.40 Continuous functions are integrable ¶
- 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 16.60 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 theorem 16.36 Euler–Lagrange equations for several independent variables ¶ ↺
      - depends_on theorem 16.38 Euler–Poisson equation ¶
      - depends_on theorem 16.41 Multiplier rule for pointwise constraints ¶
      - depends_on theorem 16.31 Natural boundary condition ¶
      - 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 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 lemma 16.18 Fundamental lemma ¶ ↺
- depends_on remark 7.128 What the derivations below take as given ¶ ↺
- 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.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.452 Young and Hölder ¶
      - depends_on lemma A.489 Courant–Lebesgue ¶
      - 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 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 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 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 lemma A.520 A $C^{1}$ limit ¶ ↺
    - depends_on lemma A.172 Dirichlet ¶
      - depends_on lemma A.193 A cosine integral ¶
      - depends_on theorem A.177 Fourier inversion for a distribution function ¶
    - 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 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 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 ¶
depends_on theorem 7.109 Leibniz integral rule ¶
- depends_on corollary 7.110 Variable limits of integration ¶ ↺
- depends_on lemma 7.116 Functions vanishing on a regular zero set ¶ ↺
- depends_on lemma A.607 A zero-free entire function is an exponential ¶
  - depends_on proposition A.609 The exponent is a quadratic polynomial ¶
    - depends_on proposition A.611 The constraint on $\Gamma$ ¶
      - depends_on proposition A.612 The dictionary between $A$ and $\Gamma$ ¶
- depends_on lemma A.599 One variable, complex coefficient ¶
  - depends_on lemma A.601 Several variables, complex symmetric matrix ¶
    - depends_on proposition A.612 The dictionary between $A$ and $\Gamma$ ¶ ↺
    - depends_on proposition A.602 The Wigner function of a Gaussian is a positive Gaussian ¶
      - depends_on proposition A.604 Non-negativity forbids a vanishing overlap ¶
- depends_on proposition A.524 Rate of decay ¶
- depends_on proposition A.521 Every derivative passes onto $g$ ¶ ↺
- depends_on theorem 7.137 Helmholtz decomposition ¶ ↺
… 1 more

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Edges

type	direction	node	provenance	where
`depends_on`	→	Sequential characterization	declared	parts/02-mathematical-methods/05-real-analysis.tex:414
`depends_on`	→	Bolzano–Weierstrass	declared	parts/02-mathematical-methods/05-real-analysis.tex:414
`depends_on`	←	Riemann–Lebesgue, continuous compactly supported case	declared	appendices/A-long-proofs.tex:9956
`depends_on`	←	Graphs and $C^{1}$ images have zero content	declared	appendices/A-long-proofs.tex:24548
`depends_on`	←	What zero content buys	declared	appendices/A-long-proofs.tex:24487
`depends_on`	←	The boundary strip is thin	declared	appendices/A-long-proofs.tex:25016
`depends_on`	←	Differentiation under the integral sign	declared	appendices/A-long-proofs.tex:4644
`depends_on`	←	Helly–Bray	declared	appendices/A-long-proofs.tex:9138
`depends_on`	←	Small chords cut off small arcs	declared	appendices/A-long-proofs.tex:23931
`depends_on`	←	The $n$-sphere is simply connected for $n \ge 2$	declared	parts/02-mathematical-methods/12-lie-groups-fibre-bundles.tex:1270
`depends_on`	←	Continuous argument along a path	declared	parts/02-mathematical-methods/04-topology.tex:318
`depends_on`	←	What the derivations below take as given	declared	parts/02-mathematical-methods/05-real-analysis.tex:4261
`depends_on`	←	Continuous functions are integrable	declared	parts/02-mathematical-methods/05-real-analysis.tex:788
`depends_on`	←	Leibniz integral rule	declared	parts/02-mathematical-methods/05-real-analysis.tex:3333
`depends_on`	←	Fejér	declared	parts/02-mathematical-methods/15-fourier-integral-transforms.tex:908
`proves`	←	ch:05-real-analysis@proof-10	declared	parts/02-mathematical-methods/05-real-analysis.tex:417