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Order and Completeness
The ordered field structure of ℝ and completeness properties.
Definitions
- Supremum (least upper bound)
- Infimum (greatest lower bound)
- Maximum
- Minimum
- Bounded above
- Bounded below
- Bounded set
- Real numbers ℝ
- Complex numbers ℂ
- Complex conjugate
- Absolute value on ℝ
- Modulus on ℂ
Axioms
Theorems
- Least upper bound theorem
- Greatest lower bound theorem
- Archimedean property of ℝ
- Density of ℚ in ℝ
- Density of ℝ \\ ℚ in ℝ
- Nested interval theorem
Lemmas
Propositions
- Uniqueness of supremum and infimum
- Basic algebraic properties of sup and inf
- Completeness equivalences
Metric Spaces and Topology
Metric spaces and point-set topology in the metric context.
Definitions
- Distance (metric)
- Metric space
- Open ball
- Closed ball
- Sphere
- Neighborhood
- Open set
- Closed set
- Interior
- Closure
- Boundary
- Limit point (accumulation point)
- Isolated point
- Derived set
- Dense subset
- Diameter
Theorems
- Open sets form a topology
- Closed sets are complements of open sets
- Sequential characterization of closure
- Sequential characterization of closed sets
Lemmas
Sequences and Series
Sequences, series, and convergence in ℝ and ℝ^k.
Definitions
- Convergent sequence
- Limit of a sequence
- Bounded sequence
- Monotone sequence
- Subsequence
- Limit superior (lim sup)
- Limit inferior (lim inf)
- Cauchy sequence
- Complete metric space
- Series
- Partial sums
- Convergent series
- Divergent series
- Absolutely convergent series
- Conditionally convergent series
- Rearrangement of a series
- Cauchy product
Theorems
- Monotone convergence theorem
- Cauchy criterion for convergence
- Bolzano–Weierstrass theorem
- Algebra of limits for sequences
- Squeeze theorem
- Absolute convergence implies convergence
- Comparison test
- Limit comparison test
- Ratio test
- Root test
- Integral test
- Cauchy condensation test
- Alternating series test
- Dirichlet test
- Abel test
- Rearrangement theorem (absolutely convergent)
- Riemann rearrangement theorem
- Mertens theorem
Lemmas
- Monotone subsequence lemma
- Basic properties of lim sup and lim inf
- Uniqueness of limits
- A convergent sequence is Cauchy
- Every Cauchy sequence is bounded
Corollaries
- Every bounded sequence in ℝ^k has a convergent subsequence
- A convergent series has terms tending to 0
Continuity
Limits and continuity of functions.
Definitions
- Limit of a function at a point
- One-sided limit
- Limit at infinity
- Continuity at a point
- Continuity on a set
- Uniform continuity
- Lipschitz continuity
- Hölder continuity
- Isometry
- Homeomorphism
Theorems
Propositions
- Equivalent definitions of continuity
- Uniform continuity implies continuity
- Uniform continuity preserves Cauchy sequences
Compactness and Connectedness
Compactness and connectedness in metric spaces.
Definitions
- Compact set
- Sequentially compact set
- Totally bounded set
- Connected set
- Separated sets
- Connected component
- Path
- Path-connected set
- Interval
- Curve
- Nowhere dense set
- Meager set
- Residual set
- Baire space
Theorems
- Sequential compactness equals compactness
- Finite intersection property theorem
- Lebesgue number lemma
- Compactness implies completeness
- Compactness implies total boundedness
- Compact iff complete and totally bounded
- Continuous image of compact set is compact
- Extreme value theorem
- Heine–Cantor theorem
- Continuous bijection from compact is homeomorphism
- Heine–Borel theorem
- Continuous image of connected set is connected
- Connected subsets of ℝ are intervals
- Intermediate value theorem
- Cantor intersection theorem
- Baire category theorem
- Banach fixed point theorem
- Arzelà–Ascoli theorem
Lemmas
Corollaries
- Continuous function on compact is bounded
- Continuous function on compact attains max and min
- Continuous function on compact is uniformly continuous
One-Variable Differentiation
Definitions
- Differentiability at a point
- Difference quotient
- Derivative
- Right/left derivative
- Higher derivatives
- Class C^k function
- Critical point
- Local maximum / local minimum
- Global maximum / global minimum
- Taylor polynomial
- Remainder term in Taylor's theorem
Theorems
- Rolle's theorem
- Mean value theorem
- Cauchy mean value theorem
- Fixed sign of derivative implies monotonicity
- Taylor's theorem with remainder
- Darboux theorem
- Inverse function theorem (1D)
- L'Hôpital's rule
Propositions
- Differentiability rules
- Derivative zero implies constant
- Bounded derivative implies uniformly continuous
Corollaries
Riemann Integration
Riemann and Riemann–Stieltjes integration.
Definitions
- Step function
- Partition of an interval
- Refinement
- Mesh of partition
- Upper sum
- Lower sum
- Tagged partition
- Riemann sum
- Riemann integrable function
- Riemann integral
- Oscillation of a function
- Set of measure zero
- Jordan content
- Riemann–Stieltjes integral
- Integrator function
Theorems
- Existence of Riemann integral for continuous functions
- Riemann integrability of monotone functions
- Riemann integrability with finitely many discontinuities
- Lebesgue criterion for Riemann integrability
- Mean value theorem for integrals
- Fundamental theorem of calculus (Part I)
- Fundamental theorem of calculus (Part II)
- Substitution rule
- Riemann–Stieltjes integrability theorem
- Integration by parts (R-S)
Lemmas
Propositions
- Riemann integrability implies boundedness
- |f| integrable if f integrable
- Closure properties (sums, products)
Corollaries
Function Sequences and Series
Sequences and series of functions, uniform convergence, power series.
Definitions
- Pointwise convergence
- Uniform convergence
- Uniform Cauchy sequence of functions
- Uniform convergence on compact sets
- Series of functions
- Equicontinuity
Theorems
- Uniform limit theorem for continuity
- Weierstrass M-test
- Uniform convergence and integration theorem
- Uniform convergence and differentiation theorem
- Dini's theorem
- Weierstrass approximation theorem
- Stone–Weierstrass theorem
- Cauchy–Hadamard theorem
- Uniform convergence of power series on compact subsets
- Term-by-term differentiation of power series
- Term-by-term integration of power series
- Abel's theorem
- Completeness of C(K)
Lemmas
- Uniform convergence implies uniform Cauchy
- Uniform Cauchy implies uniform convergence
- Uniform convergence preserves boundedness
Corollaries
- Uniform convergence implies pointwise
- Uniform limit of continuous is continuous
- Power series are analytic on disk of convergence
Multivariable Calculus
Multivariable differentiation and integration.
Definitions
- Partial derivative
- Mixed partial derivative
- Directional derivative
- Gradient
- Jacobian matrix
- Jacobian determinant
- Hessian matrix
- Total derivative (Fréchet derivative)
- Differentiable map
- Class C^k map
- Diffeomorphism
- Implicitly defined function
- Regular point / critical point
- Regular value / critical value
- Multiple integral over a rectangle
- Iterated integral
- Change of variables
- Constraint set
- Lagrange multiplier condition
Theorems
- Differentiability implies continuity
- Chain rule (multivariable)
- Mean value inequality
- C^1 implies differentiable
- Schwarz (Clairaut) theorem
- Taylor's theorem (several variables)
- Inverse function theorem (ℝ^k)
- Implicit function theorem
- Fubini theorem (Riemann)
- Change of variables formula
- Lagrange multipliers theorem
Lemmas
Corollaries
- Equality of mixed partials under C^2
- Local diffeomorphism corollary
- Implicit function parameterization corollary