Real Analysis |

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Order and Completeness

The ordered field structure of ℝ and completeness properties.

Definitions

Axioms

Theorems

Lemmas

Supremum approximation lemma

Propositions

Metric Spaces and Topology

Metric spaces and point-set topology in the metric context.

Definitions

Theorems

Lemmas

Sequences and Series

Sequences, series, and convergence in ℝ and ℝ^k.

Definitions

Theorems

Lemmas

Corollaries

Continuity

Limits and continuity of functions.

Definitions

Theorems

Propositions

Compactness and Connectedness

Compactness and connectedness in metric spaces.

Definitions

Theorems

Lemmas

Corollaries

One-Variable Differentiation

Definitions

Theorems

Propositions

Corollaries

Riemann Integration

Riemann and Riemann–Stieltjes integration.

Definitions

Theorems

Lemmas

Propositions

Corollaries

Newton–Leibniz formula

Function Sequences and Series

Sequences and series of functions, uniform convergence, power series.

Definitions

Theorems

Lemmas

Corollaries

Multivariable Calculus

Multivariable differentiation and integration.

Definitions

Theorems

Lemmas

Mean value estimate lemma

Corollaries

Additional Topics

Fixed Point Theory

Euclidean Space Structure

Function Space Topics

Definitions

Abel Test (series) — If a series converges and coefficients are bounded monotone, then the product series converges
Abel's Theorem (power series at x=1) — If a series sum a_n converges, then sum a_n r^n tends to the same limit as r→1−
Absolute convergence implies convergence — If the series of absolute values converges, then the original series converges
Absolute convergence implies the Cauchy criterion — If sum |a_n| converges then the series has arbitrarily small tails in absolute value
Absolute value on ℝ — The nonnegative magnitude |x| of a real number x, equal to its distance from 0.
Absolute value preserves Riemann integrability — If f is Riemann integrable then |f| is Riemann integrable, and |∫f| ≤ ∫|f|
Absolutely convergent series — A series ∑ a_n such that ∑ |a_n| converges.
Additivity and linearity lemmas for Riemann and Riemann–Stieltjes integrals — Linearity in the integrand and additivity over subintervals for Riemann and Riemann–Stieltjes integrals
Algebra of limits for sequences — Limits respect addition, multiplication, scalar multiplication, and (when valid) division
Algebra of Riemann integrable functions — Riemann integrable functions are closed under linear combinations and products
Algebraic properties of sup and inf — Supremum and infimum behave predictably under inclusion, translation, scaling, and unions
Alternating Series Test (Leibniz Test) — An alternating series converges if term magnitudes decrease to 0
Archimedean property of R — There are no infinitely large or infinitely small positive reals relative to the integers
Arzelà–Ascoli Theorem — On a compact metric space, equicontinuity and pointwise boundedness characterize relative compactness in C(K)
Baire Category Theorem — A complete metric space cannot be written as a countable union of nowhere dense sets
Baire space — A space where countable intersections of open dense sets are dense
Banach Fixed Point Theorem — A contraction on a complete metric space has a unique fixed point, found by iteration
Basic properties of lim sup and lim inf — Key identities and inequalities for limsup and liminf of a sequence
Bijective function — A function that is both injective and surjective.
Bolzano–Weierstrass Theorem — Every bounded sequence in R^k has a convergent subsequence
Boundary — The set of points where every neighborhood meets both the set and its complement.
Bounded above — A subset of an ordered set having at least one upper bound.
Bounded below — A subset of an ordered set having at least one lower bound.
Bounded derivative implies uniform continuity — A bounded derivative gives a Lipschitz bound, hence uniform continuity
Bounded sequence — A sequence whose values stay within some fixed finite distance of a point.
Bounded set — A set that stays within finite bounds, in an ordered set or in a metric space.
C^1 implies differentiable — If partial derivatives exist and are continuous, the map is differentiable
C^2 implies equal mixed partials — If f has continuous second partial derivatives, then mixed partials commute
Cantor Intersection Theorem — In a complete metric space, nested nonempty closed sets with diameters tending to 0 intersect in exactly one point
Cartesian product — The set of ordered pairs formed from two sets.
Cauchy Condensation Test — For decreasing nonnegative terms, convergence is equivalent to a condensed dyadic series
Cauchy criterion for convergence in R^k — A sequence in Euclidean space converges iff it is Cauchy
Cauchy implies bounded — Every Cauchy sequence in a metric space stays within a fixed ball
Cauchy Mean Value Theorem — A two-function generalization of the mean value theorem
Cauchy product of two series — The series formed by convolving coefficients of two given series.
Cauchy sequence — A sequence whose terms become arbitrarily close to each other.
Cauchy–Hadamard Theorem — Gives the radius of convergence of a power series via a limsup of nth roots
Cauchy–Schwarz inequality — The absolute inner product of two vectors is at most the product of their norms
Chain rule (multivariable) — The derivative of a composition is the composition (product) of derivatives
Change of variables (coordinate transformation) for multiple integrals — Replacing variables x by a smooth coordinate map to simplify a multiple integral
Change of variables formula for multiple integrals — Transforms an integral under a C^1 diffeomorphism via the absolute Jacobian determinant
Change of variables Jacobian corollary — A smooth bijective coordinate change preserves integrability and transforms the integral by the Jacobian
Characteristic function (indicator function) — The function that records membership in a set by 0/1 values.
Class C^k function (one-variable) — A function with continuous derivatives up to order k.
Class C^k map (ℝ^k→ℝ^m) — A map whose partial derivatives up to order k exist and are continuous.
Closed ball — The set of points within distance ≤ r of a center point in a metric space.
Closed set — A set whose complement is open in a metric space.
Closed sets are complements of open sets — A set is closed iff its complement is open; closed sets are stable under intersections
Closed subset of a compact set is compact — A closed subset of a compact space is compact
Closure — The smallest closed set containing a given set, equivalently points arbitrarily close to it.
Codomain — The target set into which a function maps.
Compact iff complete and totally bounded — In metric spaces, compactness is equivalent to completeness plus total boundedness
Compact set — A set for which every open cover has a finite subcover.
Compactness criteria in R^k — In Euclidean space, compactness is equivalent to closed-and-bounded and to sequential compactness
Compactness implies boundedness — A compact set in a metric space is contained in some finite-radius ball
Compactness implies closedness — A compact subset of a metric space contains all its limit points
Compactness implies completeness — A compact metric space is complete: every Cauchy sequence converges
Compactness implies total boundedness — A compact metric space can be covered by finitely many balls of any given radius
Compactness of graphs lemma — The graph of a continuous function over a compact domain is compact
Comparison Test (series) — Compare a series to a known convergent or divergent nonnegative series
Complement — The elements of a fixed universe that are not in the set.
Complete metric space — A metric space in which every Cauchy sequence converges to a point in the space.
Completeness axiom of R — Every nonempty set of real numbers bounded above has a least upper bound
Completeness equivalences — Several standard statements are equivalent ways to express completeness of the real numbers
Completeness of C(K) under the sup norm — On a compact metric space K, the space of continuous functions is complete in the sup metric
Complex conjugate — The map a+bi ↦ a-bi on complex numbers.
Complex numbers — Numbers of the form a+bi with i^2=-1, forming a field extending the reals.
Composition of functions — The function obtained by applying one function after another.
Composition preserves Riemann integrability — If f is Riemann integrable and g is continuous on its range, then g∘f is Riemann integrable
Conditionally convergent series — A convergent series that fails to converge absolutely.
Connected component — A maximal connected subset containing a given point.
Connected set — A set that cannot be decomposed into two disjoint nonempty relatively open pieces.
Connected subsets of R are intervals — A connected subset of the real line contains every point between any two of its points
Connectedness criteria in R — A subset of the real line is connected iff it contains all points between any two of its points
Constraint set (for optimization) — The subset of points satisfying the conditions of a constrained optimization problem
Continuity at a point — A function is continuous at x0 if f(x)→f(x0) as x→x0.
Continuity on a set — A function is continuous on E if it is continuous at every point of E.
Continuity via open sets — A function is continuous iff the preimage of every open set is open
Continuity via sequences — In metric spaces, f is continuous at x iff it preserves limits of sequences converging to x
Continuous bijection from compact is a homeomorphism criterion — A continuous bijection from a compact space to a Hausdorff space has continuous inverse
Continuous function attains max and min on a compact set — On compact domains, continuous functions achieve their extrema
Continuous function on a compact set is bounded — Continuous functions on compact domains have finite upper and lower bounds
Continuous function on a compact set is uniformly continuous — On compact domains, continuity automatically upgrades to uniform continuity
Continuous functions are Riemann integrable — Every continuous function on a closed interval has a Riemann integral
Continuous functions have the intermediate value property — A continuous function on an interval takes all values between f(a) and f(b)
Continuous functions map compact sets to closed and bounded sets in R^k — If K is compact and f is continuous into R^k, then f(K) is closed and bounded
Continuous functions on compact sets are bounded — A continuous real-valued function on a compact set has finite sup norm
Continuous image of compact set is compact — Continuous maps send compact sets to compact sets
Continuous image of connected set is connected — Continuous functions preserve connectedness
Contraction mapping — A self-map that strictly shrinks distances by a uniform factor <1
Convergence in product metric spaces — A sequence in X×Y converges iff each coordinate sequence converges
Convergent implies Cauchy — Every convergent sequence is Cauchy in any metric space
Convergent sequence — A sequence whose terms eventually become arbitrarily close to a single limit point.
Convergent series — A series whose sequence of partial sums converges in ℝ or ℂ.
Convergent series terms go to zero — If a series converges, its terms must converge to 0
Critical point — A point where the derivative is zero or fails to exist (one-variable context).
Curve (parametrized curve) — A continuous (often differentiable) map from an interval into ℝ^k.
Darboux's Theorem — Derivatives satisfy the intermediate value property even when they are not continuous
Dense set — A subset whose closure is the whole space (equivalently, it meets every nonempty open set)
Dense subset — A subset whose closure is the whole space.
Density of ℚ in ℝ — Between any two real numbers there is a rational number
Density of ℝ \ ℚ in ℝ — Between any two real numbers there is an irrational number
Derivative — The limit of the difference quotient, giving the best linear approximation at a point.
Derivative sign implies monotonicity — If a derivative is nonnegative, the function is nondecreasing (and similarly for other signs)
Derivative zero implies constant — If f' vanishes on an interval, then f is constant on that interval
Derived set — The set of all limit points of a given set in a metric space.
Determinant nonvanishing implies local invertibility lemma — Invertibility is stable under small perturbations, with a quantitative bound on the inverse
Diameter — The supremum of distances between pairs of points in a set.
Diffeomorphism — A C^1 bijection with a C^1 inverse between open subsets of Euclidean spaces
Difference quotient — The ratio (f(x)-f(a))/(x-a) measuring average rate of change from a to x.
Differentiability at a point (one-variable) — A real/complex function is differentiable at a point if its difference quotient has a limit.
Differentiability criterion via remainder estimate — Differentiability at a point is equivalent to a linear approximation with an o(||h||) error
Differentiability implies continuity — A differentiable map between Euclidean spaces is continuous at that point
Differentiability on an interval — A function is differentiable on an interval if it has a derivative at every point of the interval.
Differentiable map (ℝ^k→ℝ^m) — A map f is differentiable if it has a Fréchet derivative at every point of its domain.
Differentiation rules (one variable) — Linearity, product, quotient, and chain rules for derivatives
Dini's Theorem — On a compact space, monotone pointwise convergence of continuous functions to a continuous limit is uniform
Directional derivative — The derivative of f at a along a direction v, defined by a one-variable limit.
Dirichlet Test (series) — A series with bounded partial sums and decreasing coefficients converging to 0 converges
Distance (metric) — A function d(x,y) satisfying positivity, symmetry, and the triangle inequality.
Divergent series — A series whose partial sums do not converge.
Domain — The set of inputs on which a function is defined.
Empty set — The unique set with no elements.
Equicontinuity — A family of functions is equicontinuous if a single δ(ε) works uniformly over the family.
Equicontinuity + pointwise boundedness implies uniform boundedness on compact sets — On a compact domain, equicontinuity upgrades pointwise bounds to a global bound
Equicontinuity plus dense-set convergence implies uniform convergence on compacta — On a compact set, equicontinuity upgrades pointwise convergence on a dense set to uniform convergence
Equicontinuous family — A family of functions sharing a common continuity modulus at each point
Equivalence class — The subset of elements equivalent to a given element under an equivalence relation.
Equivalence relation — A relation that is reflexive, symmetric, and transitive.
Equivalent definitions of continuity (metric spaces) — Epsilon–delta, sequential continuity, and open-set preimages are equivalent in metric spaces
Euclidean norm — The standard length ‖x‖2 = sqrt(sum xi^2) of a vector in ℝ^k.
Euclidean space ℝ^k — The set of k-tuples of real numbers, viewed as k-dimensional space.
Every bounded infinite subset of R^k has a limit point — A bounded infinite set in Euclidean space has an accumulation point
Every bounded sequence in R^k has a convergent subsequence — A direct corollary form of the Bolzano–Weierstrass theorem
Extreme Value Theorem — A continuous real-valued function on a compact set attains its maximum and minimum
Field axioms (for R and C) — The algebraic rules for addition and multiplication that make R and C fields
Finite intersection property theorem — A space is compact iff every family of closed sets with the finite intersection property has nonempty intersection
Finite subcover lemma — A compact set has a finite subcover for every open cover
Finitely many discontinuities implies Riemann integrable — A bounded function with only finitely many discontinuities is Riemann integrable
Fixed point — A point x satisfying T(x)=x for a self-map T
Fubini's Theorem (Riemann, continuous case) — A continuous function on a rectangle can be integrated by iterated one-dimensional integrals
Function (map) — An assignment that sends each input to a unique output.
Fundamental Theorem of Calculus, Part I — Integrating a function produces an antiderivative at points of continuity
Fundamental Theorem of Calculus, Part II — If F' equals f, then the integral of f equals F(b)-F(a)
Global maximum and global minimum — A point where a function attains the largest/smallest value on its entire domain.
Gradient — The vector of first partial derivatives ∇f of a scalar-valued function.
Greatest Lower Bound Theorem — Nonempty subsets of R that are bounded below have an infimum in R
Heine–Borel Theorem — In R^k, a set is compact iff it is closed and bounded
Heine–Cantor Theorem — Continuous functions on compact metric spaces are uniformly continuous
Hessian matrix — The matrix of second partial derivatives of a scalar function f:ℝ^k→ℝ.
Higher derivatives — Iterated derivatives f^(n) obtained by differentiating repeatedly.
Hölder continuity — A quantitative continuity condition d(f(x),f(y)) ≤ C d(x,y)^α for some α∈(0,1].
Homeomorphism — A bijection that is continuous with continuous inverse.
Image (range) — The set of values a function actually attains.
Image of a compact connected set is a compact interval — A continuous real-valued map sends compact connected sets to closed intervals
Implicit Function Theorem — Solves F(x,y)=0 locally for y as a C^1 function of x when a Jacobian block is invertible
Implicitly defined function — A function described as a solution of an equation F(x,y)=0 rather than by an explicit formula
Indexed family of sets — A function assigning a set to each element of an index set.
Infimum (greatest lower bound) — The largest lower bound of a subset in an ordered set, if it exists.
Injective function — A function that never maps two distinct inputs to the same output.
Inner product on ℝ^k — The dot product ⟨x,y⟩ = sum xi yi defining angles and lengths in ℝ^k.
Integral Test — A positive decreasing series converges iff the corresponding improper integral converges
Integration by parts (Riemann integral) — A Riemann-integral identity derived from the product rule
Integration by parts (Riemann–Stieltjes) — A product rule for Riemann–Stieltjes integrals involving bounded-variation functions
Integrator function (Riemann–Stieltjes) — The function α whose increments define the weights in Riemann–Stieltjes sums.
Interchange limit and integral under uniform convergence — Uniform convergence justifies swapping a limit with a Riemann integral
Interior — The largest open set contained in a given set.
Intermediate Value Theorem — A continuous real function on an interval takes all intermediate values
Intersection — The set of elements common to all of the given sets.
Interval (in ℝ) — A subset of ℝ containing every point between any two of its points.
Inverse function — The function that undoes a bijective function.
Inverse Function Theorem (multivariable) — A C^1 map with invertible Jacobian at a point is locally a C^1 diffeomorphism
Inverse Function Theorem (one variable) — A differentiable strictly monotone function has a differentiable inverse with derivative 1/f'
Isolated point — A point of a set that has a neighborhood containing no other points of the set.
Isometry — A distance-preserving map between metric spaces.
Iterated integral — A multiple integral computed by integrating one variable at a time
Jacobian determinant — For f:ℝ^k→ℝ^k, the determinant det(J_f) controlling local volume scaling.
Jacobian matrix — The matrix of first partial derivatives of a map f:ℝ^k→ℝ^m.
Jordan content — The volume assigned to a finite union of rectangles, used as a precursor to measure.
Jordan decomposition lemma (bounded variation) — A function of bounded variation is the difference of two increasing functions
L'Hôpital's Rule — Evaluates certain indeterminate limits using the limit of a quotient of derivatives
Lagrange multiplier condition — A first-order necessary condition for constrained extrema with equality constraints
Lagrange multipliers theorem — A necessary first-order condition for constrained extrema under a regularity hypothesis
Least Upper Bound Theorem — Nonempty subsets of R that are bounded above have a supremum in R
Lebesgue criterion for Riemann integrability — A bounded function is Riemann integrable iff its discontinuities form a measure-zero set
Lebesgue Number Lemma — Every open cover of a compact metric space has a uniform radius so small balls lie in a single cover element
Lebesgue number lemma refinement lemma — On a compact set, an open cover can be refined by finitely many small balls subordinate to it
Limit Comparison Test — Two positive series behave the same if their term ratio has a positive finite limit
Limit inferior (lim inf) — For a real sequence, the limit of the tail infima, describing the minimal subsequential limit.
Limit of a function at a point — The value L that f(x) approaches as x approaches x0, defined by an ε–δ condition.
Limit of a function at infinity — The value L that f(x) approaches as x→∞, defined by an ε–M condition.
Limit of a sequence — A point x such that x_n becomes arbitrarily close to x as n→∞.
Limit point (accumulation point, cluster point) — A point x such that every neighborhood of x contains a point of the set different from x.
Limit superior (lim sup) — For a real sequence, the limit of the tail suprema, describing the maximal subsequential limit.
Linear map — A function between vector spaces preserving addition and scalar multiplication.
Linearity in the integrator (Riemann–Stieltjes) — Integrability and the integral are linear with respect to linear combinations of integrators
Lipschitz continuity — A quantitative continuity condition: |f(x)-f(y)| ≤ L|x-y| for some constant L.
Local diffeomorphism corollary — A C^1 map with invertible derivative at a point is a C^1 diffeomorphism on small neighborhoods
Local implicit function parameterization — Under the implicit function theorem hypotheses, the solution set is locally a graph
Local maximum and local minimum — A point where a function attains a maximum/minimum relative to nearby points.
Lower bound — An element that is less than or equal to every element of a given subset in an ordered set.
Lower sum (Riemann) — A weighted sum of infima of f over subintervals of a partition.
M-test continuity and integration corollary — Under the M-test, a function series converges uniformly, giving continuity and term-by-term integration
Maximum — An element of a set that is greater than or equal to every other element.
Meager set — A set that is a countable union of nowhere dense sets.
Mean value estimate lemma (differentiable maps) — Near a point where Df is continuous, f is uniformly close to its linearization
Mean value inequality (multivariable) — Bounds the change of a differentiable map by the supremum of its derivative norm
Mean Value Theorem — A theorem relating a derivative at an interior point to the average slope on an interval
Mean Value Theorem for integrals — A continuous function attains its average value over an interval
Mertens theorem on Cauchy products — Convergence of the Cauchy product under absolute convergence of one factor
Mesh of a partition — The maximum subinterval length in a partition of [a,b].
Metric space — A set equipped with a metric, used to define limits and continuity abstractly.
Minimum — An element of a set that is less than or equal to every other element.
Mixed partial derivative — An iterated partial derivative such as ∂^2f/(∂x_i∂x_j).
Modulus (absolute value) on ℂ — The nonnegative magnitude |z| of a complex number z, equal to its distance from 0.
Monotone Convergence Theorem (for sequences) — A bounded monotone sequence of real numbers converges
Monotone functions are Riemann integrable — Every bounded monotone function on a closed interval is Riemann integrable
Monotone sequence — A real sequence that is nondecreasing or nonincreasing with respect to the order.
Monotone subsequence lemma — Every real sequence has a monotone subsequence
Multiple (Riemann) integral over a rectangle — The Riemann integral of a function on a rectangle in R^n, defined via partitions and Riemann sums
Neighborhood — A set containing an open ball around a point in a metric space.
Nested Interval Theorem — A decreasing sequence of closed intervals with lengths going to 0 has a unique common point
Newton–Leibniz formula — If F is an antiderivative of f, then the integral of f equals F(b)-F(a)
Nowhere dense set — A set whose closure has empty interior.
One-sided limit — A limit as x approaches a from the left or from the right in ℝ.
Open ball — The set of points within distance < r of a center point in a metric space.
Open set — A set in a metric space that contains an open ball around each of its points.
Open sets form a topology — In a metric space, unions of open sets are open and finite intersections of open sets are open
Operator norm — The norm of a linear map defined as the maximal expansion of unit vectors.
Order axioms (for R as an ordered field) — The axioms for a total order compatible with addition and multiplication
Ordered pair — A pair (a,b) in which order matters, characterized by a uniqueness property.
Orthogonality — The relation x ⟂ y defined by ⟨x,y⟩ = 0 in an inner product space.
Oscillation criterion lemma — Upper minus lower sum equals the oscillation sum, yielding a practical integrability criterion
Oscillation of a function — The quantity sup f − inf f on a set, measuring variation in values.
Partial derivative — The derivative with respect to one coordinate of a function f:ℝ^k→ℝ^m.
Partial order — A relation that is reflexive, antisymmetric, and transitive.
Partial sums — The finite sums s_N = ∑_{n=1}^N a_n associated to a series.
Partition — A decomposition of a set into disjoint nonempty pieces covering the whole set.
Partition of an interval — A finite increasing list a=x0<⋯<xn=b subdividing [a,b] into subintervals.
Path — A continuous map γ:[a,b]→X, used to connect points in a space.
Path-connected set — A set in which any two points can be joined by a continuous path.
Pointwise bounded family — A family of functions whose values are bounded at each fixed point of the domain
Pointwise convergence — A sequence of functions f_n converges pointwise if f_n(x)→f(x) for each x.
Positive derivative implies strictly increasing — If f' is positive everywhere on an interval, f is strictly increasing
Power series are analytic on their disk of convergence — Inside the radius of convergence, a power series can be differentiated term-by-term indefinitely
Power set — The set of all subsets of a given set.
Preimage (inverse image) — The set of inputs that a function maps into a given subset of the codomain.
Principle of mathematical induction — An axiom scheme asserting that properties holding at 1 and preserved by n→n+1 hold for all natural numbers
Proper subset — A subset that is strictly smaller than the containing set.
Ratio Test — A series converges absolutely if successive term ratios are eventually less than 1
Real numbers (as a complete ordered field) — A field with a compatible total order and the least upper bound property.
Rearrangement of a series — A permutation of the terms of a series, defined via a bijection of ℕ.
Rearrangement theorem for absolutely convergent series — Reordering an absolutely convergent series does not change its sum
Refinement lemma for upper and lower sums — Refining a partition increases lower sums and decreases upper sums
Refinement of a partition — A partition Q that contains all points of a partition P (possibly with extra points).
Regular point and critical point — Points where the derivative of a map has maximal rank, versus points where it fails to
Regular value and critical value — Values whose preimages contain only regular points, versus values hit at some critical point
Relation — A subset of a Cartesian product, viewed as a set of ordered pairs.
Relatively compact set — A subset whose closure is compact (also called precompact)
Remainder term in Taylor's theorem — The difference f(x)−T_k f(x;a), measuring Taylor approximation error.
Residual set — A set whose complement is meager.
Restriction of a function — The same function viewed only on a specified subset of its domain.
Reverse triangle inequality — The difference of norms is bounded by the norm of the difference
Riemann integrability implies boundedness — A Riemann integrable function on a closed interval must be bounded
Riemann integrable function — A bounded function on [a,b] for which upper and lower sums can be made arbitrarily close.
Riemann integral — The common value of the upper and lower integrals of a Riemann integrable function.
Riemann rearrangement theorem — A conditionally convergent real series can be rearranged to converge to any prescribed value or to diverge
Riemann sum — A finite sum ∑ f(t_i)Δx_i associated to a tagged partition of [a,b].
Riemann–Stieltjes integrability theorem — A continuous integrand is Riemann–Stieltjes integrable against an increasing integrator
Riemann–Stieltjes integral — An integral ∫ f dα defined via limits of sums using increments of an integrator α.
Right derivative and left derivative — One-sided derivatives defined by one-sided limits of the difference quotient.
Rolle's Theorem — If a differentiable function agrees at the endpoints, it has a critical point inside
Root Test — A series converges absolutely if the nth roots of term magnitudes have limsup less than 1
Schwarz's Theorem (Clairaut's theorem) — Under continuity of second partials, mixed partial derivatives are equal
Second derivative tests — Using second derivatives (or the Hessian) to classify critical points as local minima or maxima
Separated sets — Two sets A,B are separated if neither meets the closure of the other.
Sequential characterization of closed sets — In metric spaces, a set is closed iff it contains limits of convergent sequences from itself
Sequential characterization of closure — In metric spaces, x is in the closure of E iff some sequence in E converges to x
Sequential compactness equals compactness (metric spaces) — In metric spaces, compactness is equivalent to every sequence having a convergent subsequence
Sequentially compact set — A set in which every sequence has a convergent subsequence with limit in the set.
Series (summable family) — An infinite sum ∑ a_n defined via convergence of its partial sums.
Series of functions — An infinite sum ∑ f_n defined via convergence of partial sums as functions.
Set — A primitive object for which membership is defined.
Set difference — The subset of one set obtained by removing elements belonging to another.
Set of measure zero in ℝ^k — A set that can be covered by countably many rectangles (or balls) with arbitrarily small total volume.
Sphere (metric sphere) — The set of points at distance exactly r from a center point in a metric space.
Squeeze Theorem — If a sequence or function is trapped between two that share a limit, it has that limit
Step function (on an interval) — A function constant on each subinterval of a finite partition.
Stone–Weierstrass Theorem — A subalgebra of continuous functions that separates points is dense in C(K)
Subsequence — A sequence obtained by selecting terms along a strictly increasing index sequence.
Subset — A set contained in another set, defined by elementwise inclusion.
Substitution rule (change of variables) for Riemann integrals — A one-dimensional change of variables formula for definite integrals
Supremum (least upper bound) — The smallest upper bound of a subset in an ordered set, if it exists.
Supremum approximation lemma — A supremum can be approached from below by points in the set
Surjective function — A function whose image equals its codomain.
Symmetric difference — The set of elements that belong to exactly one of two sets.
Tagged partition — A partition together with a chosen sample point in each subinterval.
Taylor polynomial — The polynomial formed from the first k derivatives of f at a point a.
Taylor's Theorem in several variables — Approximates a smooth multivariable function by a polynomial in a neighborhood of a point
Taylor's Theorem with remainder — Approximates a smooth function by a polynomial with a controlled error term
Term-by-term differentiation of power series — Inside the radius of convergence, a power series can be differentiated term-by-term
Term-by-term integration of power series — Inside the radius of convergence, a power series can be integrated term-by-term
Term-by-term operations on series of functions — Uniform convergence hypotheses justify integrating or differentiating a function series term-by-term
Total boundedness characterization via ε-nets — A set is totally bounded iff it has a finite ε-net for every ε>0
Total derivative (Fréchet derivative in ℝ^k) — The linear map Df(a) giving the best first-order approximation f(a+h)=f(a)+Df(a)h+o(‖h‖).
Total order (linear order) — A partial order in which every pair of elements is comparable.
Totally bounded iff every sequence has a Cauchy subsequence — A metric set admits finite ε-nets for all ε>0 exactly when sequences have Cauchy subsequences
Totally bounded set — A set that can be covered by finitely many ε-balls for every ε>0.
Triangle inequality — Distances and norms satisfy a subadditivity inequality
Uniform Cauchy implies uniform convergence — In a complete codomain, uniformly Cauchy function sequences converge uniformly
Uniform Cauchy sequence of functions — A sequence (f_n) such that sup_x d(f_m(x),f_n(x))→0 as m,n→∞.
Uniform continuity — Continuity with a single δ(ε) working uniformly for all points in the domain.
Uniform continuity implies continuity — Uniform continuity gives a single delta that works at every point, hence pointwise continuity
Uniform continuity preserves Cauchy sequences — Uniformly continuous maps send Cauchy sequences to Cauchy sequences
Uniform convergence (sequence of functions) — Convergence f_n→f with a single N(ε) working for all x in the domain.
Uniform convergence (series of functions) — A series ∑ f_n converges uniformly if its partial sums converge uniformly.
Uniform convergence and differentiation — Uniform convergence of derivatives plus convergence at one point implies uniform convergence of functions and term-by-term differentiation
Uniform convergence and integration — Uniform limits of integrable functions are integrable and integrals commute with uniform limits
Uniform convergence and sup norm — Uniform convergence is exactly convergence in the sup norm, and sup norms are Lipschitz under it
Uniform convergence implies pointwise convergence — Uniform convergence is stronger than pointwise convergence
Uniform convergence implies uniform Cauchy — A uniformly convergent sequence of functions is uniformly Cauchy
Uniform convergence of power series on compact subsets — A power series converges uniformly on closed balls strictly inside its radius of convergence
Uniform convergence on compact sets — Convergence that is uniform when restricted to each compact subset of the domain.
Uniform convergence preserves boundedness — If functions converge uniformly and one is bounded, then all later ones and the limit are bounded
Uniform limit of continuous functions is continuous — A corollary of the uniform limit theorem for continuity
Uniform limit of integrable functions is integrable — Uniform convergence preserves Riemann integrability and allows exchanging limit and integral
Uniform limit theorem for continuity — A uniform limit of continuous functions is continuous
Uniformly bounded family — A family of functions bounded by a single constant on the whole domain
Union — The set of elements that belong to at least one of the given sets.
Uniqueness of limits — A convergent sequence in a metric space has only one limit
Uniqueness of supremum and infimum — A set has at most one least upper bound and at most one greatest lower bound
Upper bound — An element that is greater than or equal to every element of a given subset in an ordered set.
Upper sum (Riemann) — A weighted sum of suprema of f over subintervals of a partition.
Weierstrass Approximation Theorem — Polynomials are dense in the space of continuous functions on a closed interval
Weierstrass M-test — A comparison test guaranteeing uniform convergence of a series of functions
Well-ordering principle for N — Every nonempty subset of the natural numbers has a least element
Zero derivative implies constant — If f' vanishes everywhere on an interval, the function is constant