Section
Real Analysis
Sequences, series, continuity, differentiation, and integration on the real line
Core idea
Click any term to expand its definition inline.
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Order and Completeness
Definitions
Axioms
Theorems
- Archimedean property
- Density of ℚ
- Density of irrationals
- Completeness equivalences
- Supremum approximation lemma
- Uniqueness of sup and inf
- Sup/inf algebra
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Sequences and Series
Sequences
Sequence Theorems
Series Definitions
- Series
- Partial sums
- Convergent series
- Divergent series
- Absolutely convergent series
- Conditionally convergent series
- Rearrangement of series
- Cauchy product
- Power series
Convergence Tests
- Terms go to zero
- Comparison test
- Limit comparison test
- Ratio test
- Root test
- Integral test
- Cauchy condensation test
- Alternating series test
- Dirichlet test
- Abel test
Series Theorems
- Absolute ⟹ convergent
- Absolute ⟹ Cauchy
- Rearrangement (absolute)
- Riemann rearrangement theorem
- Mertens theorem
- Cauchy–Hadamard theorem
- Abel's theorem
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Limits and Continuity
Limits
Monotone Functions
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Differentiation
One-Variable Definitions
- Derivative
- Differentiability (1D)
- Higher derivatives
- Class Cᵏ function
- Critical point
- Local extremum
- Global extrema
- Taylor polynomial
Differentiation Rules
Mean Value Theorems
- Rolle's theorem
- Mean value theorem
- Cauchy mean value theorem
- Mean value estimate
- Mean value inequality
Consequences
- f′ = 0 ⟹ constant
- Sign of f′ ⟹ monotonicity
- f′ > 0 ⟹ increasing
- Bounded derivative ⟹ uniform continuity
- Darboux theorem
- L'Hôpital's rule
- Inverse function theorem (1D)
- Taylor's theorem
- Second derivative tests
- Intermediate value theorem
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Multivariable Differentiation
Definitions
- Partial derivative
- Mixed partial derivative
- Directional derivative
- Gradient
- Jacobian matrix
- Jacobian determinant
- Hessian matrix
- Fréchet derivative
- Differentiable map
- Class Cᵏ map
- Diffeomorphism
Implicit/Inverse Functions
- Implicitly defined function
- Implicit function theorem
- Inverse function theorem (ℝᵏ)
- Local diffeomorphism corollary
- Implicit function parameterization
Critical Points
- Regular point
- Regular value
- Critical value
- Constraint set
- Lagrange multiplier condition
- Lagrange multipliers theorem
Theorems
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Riemann Integration
Definitions
- Partition
- Refinement
- Mesh
- Tagged partition
- Lower sum
- Upper sum
- Riemann sum
- Riemann integrable function
- Riemann integral
- Step function
- Oscillation
Basic Properties
- Integrability ⟹ bounded
- Linearity
- Algebra of integrable functions
- Refinement lemma
- |f| integrable
- Composition preserves integrability
Integrability Criteria
- Continuous ⟹ integrable
- Monotone ⟹ integrable
- Finite discontinuities ⟹ integrable
- Oscillation criterion
Fundamental Theorems
Multiple Integrals
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Riemann–Stieltjes Integration
- Integrator function
- Riemann–Stieltjes integral
- R-S integrability theorem
- R-S linearity
- Integration by parts (R-S)
- Bounded variation
- Total variation
- Jordan decomposition
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Function Sequences and Series
Convergence Modes
Uniform Convergence Properties
- Uniform ⟹ pointwise
- Uniform Cauchy ⟺ uniform
- Uniform preserves boundedness
- Uniform ⟺ sup-norm
- Uniform limit of continuous
- Uniform limit of integrable
- Weierstrass M-test
- M-test corollary
Interchange Theorems
- Interchange limit and integral
- Uniform convergence and integration
- Uniform convergence and differentiation
- Dini's theorem
Power Series
- Power series uniform on compacts
- Power series analytic on disk
- Term-by-term differentiation
- Term-by-term integration
- Term-by-term operations
Approximation
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Function Spaces
- Space of continuous functions
- Supremum norm
- Uniform metric
- Equicontinuity
- Equicontinuous family
- Pointwise bounded family
- Uniformly bounded family
- Arzelà–Ascoli theorem
- Equicontinuity boundedness criterion
- Equicontinuity dense set lemma
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Cauchy Criterion
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Uncategorized
- Additivity and linearity lemmas for Riemann and Riemann–Stieltjes integrals
- Algebraic properties of sup and inf
- Banach Fixed Point Theorem
- Basic properties of lim sup and lim inf
- Every bounded sequence in R^k has a convergent subsequence
- C^2 implies equal mixed partials
- Chain rule (multivariable)
- Continuity via sequences
- Convergent series terms go to zero
- Density of in
- Difference quotient
- Equicontinuity + pointwise boundedness implies uniform boundedness on compact sets
- Equivalent definitions of continuity (metric spaces)
- Continuous functions are Riemann integrable
- Finite subcover lemma
- Fixed point
- Global maximum and global minimum
- Greatest Lower Bound Theorem
- Image (range)
- Isolated point
- Least Upper Bound Theorem
- Limit of a function at a point
- Limit of a sequence
- Limit point (accumulation point, cluster point)
- Local maximum and local minimum
- Lower sum (Riemann)
- M-test continuity and integration corollary
- Mertens theorem on Cauchy products
- Modulus (absolute value) on ℂ
- Determinant nonvanishing implies local invertibility lemma
- Newton–Leibniz formula
- Positive derivative implies increasing
- Preimage (inverse image)
- Regular point and critical point
- Regular value and critical value
- Remainder term in Taylor
- Reverse triangle inequality
- Right derivative and left derivative
- Substitution rule (change of variables) for Riemann integrals
- Taylor
- Total boundedness characterization via ε-nets
- Total derivative (Fréchet derivative in ℝ^k)
- Uniform continuity preserves Cauchy sequences
- Uniform convergence (sequence of functions)
- Uniform limit theorem for continuity
- Upper sum (Riemann)
- Zero derivative implies constant
Coordinates and calculus additions
- Cartesian coordinates
- Cosine function
- Curl of a three-dimensional vector field
- A sufficient criterion for smoothness at a cylindrical axis
- Cylindrical coordinates and their orthonormal frame
- Divergence of a Euclidean vector field
- Row divergence of a matrix field
- Vector field on a Euclidean open set
- Exponential function
- Euclidean Laplacian
- Logarithmic radial derivative
- Natural logarithm
- Nonnegative square root
- Product rule for derivatives
- Quotient rule
- Real power of a positive number
- Sine function
- Stokes streamfunction for an axisymmetric meridional field
- Matrix-valued tensor field in Euclidean coordinates
- Vector potential for a divergence-free field
Further integration prerequisites
Analytic dependence and coefficient estimates
Smooth localization and extension
- compactly supported function
- cutoff function
- cartesian jet
- flat function
- multi index leibniz rule
- flat exponential
- smooth extension
- smooth zero extension
- hadamard division lemma
- flat factor product
- normalized flat primitive
- scaled cutoff estimate
- compact derivative seminorm
- squared partition of unity
- solenoidal localization
- borel jet extension
Endpoint bounds for evolution equations
Scaling and weighted differential estimates
- Anisotropic dilation
- Norms of an anisotropically concentrating profile
- Smooth dyadic partition of a positive scale
- Implicit anisotropic similarity coordinates
- Chain rule in implicit anisotropic coordinates
- Normalized dyadic chart
- Scaling of weighted radial moments
- Linearization of a nonlinear map
- Flat logarithmic interval weight
- Weighted coefficient class
- Envelope-controlled amplitude
- Product bounds for weighted coefficients and amplitudes
Periodicity
Torus covers and support combinatorics
- Chain rule for iterated torus variables
- Irrational number
- Quadratic algebraic conjugate
- Quadratic irrational
- Quadratic irrational lower bound
Cones, averages and moment constraints
- Inverse loss when two moment exponents coalesce
- Cumulative weighted integral
- Invertibility of moments on ordered bump supports
- Matching prescribed integral moments
- Positive powers of flat weighted profiles
- Preservation of exterior primitives by moment matching
- Circle reparametrization by a positive density
- Compactly supported weighted radial primitive
- Projection removing one weighted moment
- Weighted radial moment
- Zero count for distinct real powers
Heat flow and integral profiles
- Gamma average of an algebraic factor
- Gamma function on positive real arguments
- Gaussian integral
- The constant pi
- Radial Laplacian
- Smooth nonanalytic endpoint of a Gamma average
- Tricomi U for positive real arguments