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Introduction to real analysis

Learn how numbers, limits and functions work. Build up to sequences, series, derivatives and integrals, one idea at a time.

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Mathematics / University introductory course

187small concepts
20 subtopics · your own pace
SUBTOPIC 01 · 8 concepts

Foundations: sets and notation

Sets, inclusion and the notation used by later arguments.

  • Sets, elements and set-builder notation ▶
    Interpret a set as a collection determined by membership; read ∈, ∉ and set-builder notation. Distinguish x from {x}, and the empty set from {∅}; order and repetitions do not change a set. Recognize N, Z, Q and R, taking N={1,2,...} in this course.
  • Subsets, supersets and proper inclusion ▶
    Interpret A⊆B as every element of A belonging to B and equivalently B⊇A. Proper inclusion A⊊B additionally requires an element in B outside A. Explain A⊆A and ∅⊆A, including why ∅⊆A does not imply ∅∈A.
  • Proving equality by two inclusions ▶
    Prove A=B by proving both A⊆B and B⊆A. Establish transitivity of inclusion by following an arbitrary element. Sharing a common superset does not establish equality.
  • Union and intersection through membership ▶
    Determine membership in A∪B using inclusive or and in A∩B using and; distinguish an empty intersection from an empty union. Explain A∩B⊆A⊆A∪B.
  • Complements, difference and De Morgan’s laws ▶
    Interpret A∖B and the complement U∖A relative to a stated universe U. Justify the two De Morgan laws by translating membership into not, and and or.
  • Unions and intersections of a family ▶
    Interpret membership in an indexed union as belonging to at least one member and membership in an indexed intersection as belonging to every member. Treat the family and its member sets as different objects.
  • Ordered pairs and Cartesian products ▶
    Form A×B as the set of ordered pairs (a,b) with a∈A and b∈B. Pair order matters and equality is componentwise; recognize R² as pairs of real numbers.
  • Intervals and endpoints on the real line ▶
    Translate open, closed, half-open and unbounded interval notation into inequalities. Inclusion of an endpoint differs from approaching it, and infinity is not a real endpoint that can belong to an interval.
SUBTOPIC 02 · 9 concepts

Foundations: logic and proof

Read claims precisely and justify elementary results.

  • Implications, converses and equivalence ▶
    Interpret P⇒Q, its converse Q⇒P and P⇔Q. P is sufficient for Q and Q necessary for P. An implication fails precisely when its hypothesis is true and conclusion false; a false hypothesis does not refute it.
  • Quantifier order and dependent choices ▶
    Interpret ∀ and ∃ with explicit domains. In ∀x∃y the witness may depend on x; in ∃y∀x the same witness must work for every x. Distinguish a bound variable from a freely chosen parameter.
  • Negating logical and quantified claims ▶
    Negate a statement using not(P and Q)=(not P or not Q), not(P or Q)=(not P and not Q), not(P⇒Q)=(P and not Q), and exchange ∀ with ∃ while retaining domains. Negate strict and non-strict inequalities correctly.
  • Proving from a hypothesis ▶
    Prove P⇒Q for an arbitrary admissible object by using P and justified definitions, without assuming Q. Examples illustrate but do not prove a universal assertion.
  • Disproving a universal claim ▶
    Disprove ∀x∈D P(x) with one admissible witness for which P fails; explain why an example outside D is invalid and many successful examples do not establish a theorem.
  • Proving the contrapositive ▶
    Prove P⇒Q by proving not Q⇒not P; explain why these are equivalent while the converse need not be.
  • Proof by contradiction ▶
    Prove a claim by assuming its negation and deriving an impossibility from stated assumptions. For an implication, assume both the hypothesis and the negation of the conclusion.
  • Base case, hypothesis and induction step ▶
    Prove a statement for all integers n≥n0 by verifying the starting case and proving P(n)⇒P(n+1) for arbitrary n≥n0. Distinguish assuming one induction hypothesis from assuming the theorem for every n.
  • Reading and splitting a finite sum ▶
    Read summation indices as bound variables, expand a finite sum, and split off the last term: ∑_{k=1}^{n+1}a_k=∑_{k=1}^{n}a_k+a_{n+1}. Renaming the index does not change the sum.
SUBTOPIC 03 · 5 concepts

Foundations: functions and algebraic structure

Functions, set mappings and the small amount of algebra needed for ordered fields.

  • Functions, domains, codomains and images ▶
    A function f:A→B assigns exactly one output in B to every input in A. Distinguish the codomain from the image f(A); a rule can fail by missing an input or assigning two outputs.
  • Images and preimages of sets ▶
    Compute f(S) by following inputs forward and f⁻¹(T) by collecting every input mapped into T. A preimage exists as a set even when f has no inverse function.
  • Binary operations and closure ▶
    Recognize a binary operation on S as a single-valued function S×S→S, defined for every ordered pair. Distinguish closure from associativity and commutativity.
  • Identities, inverses and groups ▶
    Explain closure, associativity, a common two-sided identity and an inverse for each element; an abelian group also requires commutativity. An inverse is relative to the specified operation.
  • Two group structures and distributivity ▶
    Explain a field as an abelian additive group and an abelian multiplicative group on the nonzero elements, linked by distributivity, with 0≠1. Distinguish additive zero from multiplicative one; zero has no multiplicative inverse.
SUBTOPIC 04 · 8 concepts

Counting infinite sets

Finite and infinite cardinality, inverse mappings and diagonal arguments.

  • Bijections pair elements exactly ▶
    Distinguish injectivity, surjectivity and bijectivity for a specified domain and codomain. A function has a two-sided inverse exactly when it is bijective; changing the codomain can change surjectivity.
  • The set of all subsets ▶
    Construct P(A), including ∅ and A, and distinguish an element of A from a subset of A and an element of P(A). For a finite n-element set justify that P(A) has 2^n elements by independent include/exclude choices.
  • Countability means an enumeration ▶
    Distinguish finite, countably infinite and at-most-countable sets. Give an enumeration or bijection with N; an infinite set may have the same cardinality as a proper subset.
  • Subsets of countable sets ▶
    Prove every subset of an at-most-countable set is at most countable; for an infinite subset select its members in increasing order of their enumeration indices.
  • Listing a countable family of countable sets ▶
    Given an enumeration for each member of a countable family, enumerate their union by diagonal traversal of pairs of indices, discarding repeats; finite or empty members cause no difficulty.
  • Cantor’s diagonal subset ▶
    Prove Cantor’s theorem: no function A→P(A) is onto, using D={a∈A:a∉f(a)}. Explain why both possible membership answers at a proposed preimage of D contradict its definition.
  • The real numbers cannot be listed ▶
    Prove (0,1), hence R, is uncountable by a diagonal decimal construction. Choose digits avoiding 0 and 9 so dual terminating expansions cannot invalidate the differing-digit argument.
  • The rationals are countable ▶
    Prove Q is countable by enumerating integer-numerator, positive-denominator pairs in finite diagonal stages and removing repeated rational values. An enumeration must reach every value after finitely many stages.
SUBTOPIC 05 · 15 concepts

Real numbers and completeness

The ordered real field, bounds and completeness, followed by density and distance estimates.

  • Field operations compatible with order ▶
    State a total order compatible with field addition and multiplication of positive elements. Derive sign rules including positivity of nonzero squares; being an ordered field does not imply completeness.
  • Bounds and attained maxima ▶
    Distinguish upper/lower bounds from maximum/minimum, and bounded above/below from bounded on both sides. An extremum must belong to the set, whereas a bound need not.
  • Absolute value as distance ▶
    Interpret absolute value by its piecewise definition and as real-line distance. Translate strict and non-strict distance inequalities into interval conditions, treating endpoints correctly.
  • The least upper bound ▶
    Characterize s=sup A for a nonempty bounded-above A by the upper-bound condition and ∀ε>0 ∃a∈A with s−ε<a. Prove uniqueness and explain that a supremum need not be attained.
  • The triangle inequality ▶
    Prove |x+y|≤|x|+|y| from −|x|≤x≤|x|. Use it as a distance estimate without replacing the right side by an unjustified equality.
  • Completeness of the real line ▶
    State the least-upper-bound axiom for every nonempty subset of R bounded above. Distinguish an axiom guaranteeing a real supremum from computing that supremum or asserting attainment.
  • The greatest lower bound ▶
    Characterize inf A for nonempty bounded-below A by its lower-bound and approximation-from-above properties. Derive inf A=−sup(−A), and distinguish the infimum from an attained minimum.
  • The reverse triangle inequality ▶
    Derive ||x|−|y||≤|x−y| by applying the triangle inequality in both directions; interpret it as stability of distance from the origin.
  • The Archimedean property ▶
    Prove N is unbounded above from completeness, and deduce that for every ε>0 there is n∈N with 1/n<ε. Make strict inequalities in the chosen n explicit.
  • Existence of the real square root of two ▶
    Use completeness to establish a positive real s with s²=2 by taking the supremum of {x≥0:x²<2} and ruling out s²<2 and s²>2 with small perturbations.
  • Rationals between real numbers ▶
    Prove that between any real a<b lies a rational, using an integer n with n(b−a)>1 and an integer m with na<m≤na+1. Justify the integer selection using the Archimedean property and well-ordering of N.
  • Nested closed intervals and completeness ▶
    Prove that nested nonempty closed bounded intervals have a common point using the supremum of their left endpoints. If their lengths approach zero, the common point is unique.
  • Irrationality of the square root of two ▶
    Prove the positive real square root of two is irrational: a lowest-terms rational representation forces both numerator and denominator to be even.
  • Irrationals between real numbers ▶
    Prove every nonempty open real interval contains an irrational by translating the interval by a fixed irrational and applying rational density. Explain why adding a rational to an irrational cannot be rational.
  • The rational field is incomplete ▶
    Show that the nonempty rational set {q≥0:q²<2} has no supremum in Q, although it has one in R. Distinguish boundedness in an ordered field from the field’s completeness.
SUBTOPIC 06 · 14 concepts

Sequences and convergence

Sequence and subsequence notation, quantified limits, basic convergence estimates, sum/product laws and divergence to infinity.

  • Sequences and tails ▶
    Define a real sequence as a function N→R and a tail as its restriction to sufficiently large indices. Values may repeat, and changing finitely many terms leaves all sufficiently late terms unchanged.
  • Subsequence indices must increase ▶
    Define a subsequence a_{n_k} through strictly increasing natural indices. Prove n_k≥k and distinguish deleting terms from reordering them.
  • The epsilon–N definition ▶
    State ∀ε>0 ∃N∈N ∀n≥N, |a_n−L|<ε, with N depending on ε but not on n. Recognize finite initial changes as irrelevant to convergence.
  • The squeeze theorem ▶
    Prove the squeeze theorem for eventual inequalities and common finite bounding limit; finite exceptional indices do not matter. A bound on absolute error is a useful special case.
  • Subsequences and limits ▶
    Prove every subsequence of a convergent sequence has the same limit, using n_k≥k. Deduce that subsequences with distinct limits disprove convergence of the original sequence.
  • Uniqueness of a sequence limit ▶
    Prove uniqueness of a finite real-sequence limit using a common tail and the triangle inequality, choosing epsilon in terms of the distance between two alleged limits.
  • Convergence implies boundedness ▶
    Prove every convergent real sequence is bounded by combining one bounded tail with finitely many initial terms. The converse is false.
  • Proving the sum limit law ▶
    Prove a_n+b_n→a+b when a_n→a and b_n→b, using two epsilon/2 estimates and a common threshold. Extend to multiplication by a fixed scalar, handling zero separately.
  • Inequalities survive limits, strictness may not ▶
    Prove that a_n≤b_n eventually and a_n→a, b_n→b imply a≤b, by contradiction with a common small error bound. Strict inequalities between all terms need not give a strict inequality between the limits.
  • Proving the product limit law ▶
    Prove a_n b_n→ab for convergent real sequences by writing a_n(b_n−b)+b(a_n−a) and bounding a_n uniformly. A product of two small errors alone is not the full difference.
  • Proving a reciprocal sequence converges ▶
    Prove 1/n→0 by constructing an epsilon-dependent natural-number threshold using the Archimedean property, then verifying the inequality for every later index.
  • Divergence to infinity ▶
    Define a_n→+∞ by eventual exceedance of every real bound, with the analogous definition for −∞. Unboundedness alone does not imply either behavior.
  • Algebra of sequence limits ▶
    Apply linear-combination, product and quotient limit laws for convergent real sequences. A nonzero denominator limit guarantees denominators are eventually nonzero and bounded away from zero; a zero denominator limit gives no general quotient conclusion.
  • Geometric powers tend to zero ▶
    Prove r^n→0 for |r|<1: for 0<|r|<1, the positive powers decrease to L≥0, and the shifted sequence and product law give L=|r|L, hence L=0. Handle r=0 and negative r explicitly.
SUBTOPIC 07 · 9 concepts

Sequence compactness and completeness

Monotonicity, subsequence selection, Bolzano–Weierstrass and Cauchy completeness, building on the basic limit laws.

  • Monotonicity and boundedness are different ▶
    Distinguish nondecreasing/nonincreasing from strictly increasing/decreasing, and each from boundedness. Test monotonicity using consecutive terms while keeping signs and index domains explicit.
  • Cauchy sequences ▶
    State the two-index Cauchy condition with one threshold valid for all m,n beyond it, without presupposing a limit. Negate it with a fixed positive tolerance and arbitrarily late separated pairs.
  • A monotone subsequence of every real sequence ▶
    Prove every real sequence admits a monotone subsequence by dividing into infinitely many peak indices and finitely many peak indices. Select strictly increasing indices in either construction.
  • A convergent sequence is Cauchy ▶
    Prove a convergent real sequence is Cauchy by placing both late terms within ε/2 of its limit and applying the triangle inequality.
  • A Cauchy sequence is bounded ▶
    Prove a Cauchy sequence is bounded by comparing all sufficiently late terms to one fixed tail term and then bounding the finite prefix.
  • Bounded monotone convergence ▶
    Prove bounded nondecreasing real sequences converge to the supremum of their ranges; prove the nonincreasing version by negation. Monotonicity without the relevant one-sided bound need not give a finite limit.
  • One convergent subsequence controls a Cauchy sequence ▶
    Prove a Cauchy sequence having a subsequence converging to L must itself converge to L by a two-error estimate using one sufficiently late subsequence term.
  • Bolzano–Weierstrass ▶
    Prove every bounded real sequence has a convergent subsequence by monotone subsequence selection and bounded monotone convergence. The theorem does not assert convergence of the entire sequence.
  • The Cauchy convergence criterion ▶
    Establish that a real sequence converges if and only if it is Cauchy, linking boundedness, Bolzano–Weierstrass and the subsequence-limit lemma. Explain that the implication from Cauchy to convergence depends on completeness of the ambient number system.
SUBTOPIC 08 · 5 concepts

Subsequential limits and oscillation

Tail bounds, extreme subsequential limits and oscillation.

  • Upper and lower bounds of a tail ▶
    For bounded real sequences define s_n=sup{a_k:k≥n} and i_n=inf{a_k:k≥n}. Nested tails make s_n nonincreasing and i_n nondecreasing, and i_n≤a_n≤s_n.
  • Limsup and liminf ▶
    Define limsup a_n as the limit of tail suprema and liminf a_n as the limit of tail infima for bounded sequences; monotone convergence ensures both exist. Distinguish extreme tail behavior from the maximum/minimum of the entire range.
  • When upper and lower limits agree ▶
    Prove a bounded real sequence converges to L exactly when liminf and limsup are both L, using the tail-bound squeeze. Always liminf≤limsup.
  • Largest and smallest subsequential limits ▶
    For a bounded real sequence prove that its limsup and liminf occur as subsequential limits and that every subsequential limit lies between them. Intermediate values need not occur as subsequential limits.
  • Infinite upper and lower limits ▶
    Extend limsup and liminf using extended-real tail bounds; +∞ and −∞ describe behavior rather than ordinary real values for unrestricted arithmetic. Characterize convergence to +∞ by liminf=+∞, and to −∞ by limsup=−∞.
SUBTOPIC 09 · 11 concepts

Infinite series

Partial sums, Cauchy tails, convergence tests and power-series radii.

  • A series is a sequence of partial sums ▶
    Define ∑a_n through its sequence of partial sums, distinguishing a term from a partial sum and a finite sum from its infinite-series limit. Changing finitely many terms preserves convergence but generally changes the sum.
  • Divergence of the harmonic series ▶
    Prove divergence of ∑1/n by grouping positive terms into dyadic blocks, each contributing at least 1/2. Terms tending to zero are not sufficient for convergence.
  • Comparison of positive series ▶
    For eventually nonnegative terms with 0≤a_n≤b_n, prove that convergence of ∑b_n forces convergence of ∑a_n and divergence of ∑a_n forces divergence of ∑b_n. The reversed conclusions are invalid.
  • Alternating series and their remainder ▶
    Prove the alternating-series theorem for magnitudes decreasing to zero using monotone even/odd partial sums and their common limit. The remainder after N terms has magnitude at most the next term and the sign of the first omitted term unless zero.
  • The necessary term test ▶
    Derive a_n→0 from convergence of the partial sums by a_n=S_n−S_{n−1}. If terms fail to approach zero the series diverges, but the converse is false.
  • Cauchy tails of a series ▶
    Prove ∑a_n converges in R exactly when for each ε>0 all sufficiently late finite tails |∑_{k=m}^n a_k| are below ε for n≥m. Translate between tail sums and differences of partial sums.
  • Geometric series ▶
    Derive finite geometric sums and obtain ∑_{n=0}∞ar^n=a/(1−r) for |r|<1. For nonzero a and |r|≥1 the terms do not approach zero; a=0 is a separate trivial case.
  • The ratio test ▶
    State the ratio test for eventually nonzero terms and L=lim |a_{n+1}/a_n|: absolute convergence if L<1 and divergence if L>1, including +∞; L=1 is inconclusive. Recognize when the ratio is undefined.
  • The root test ▶
    State the root test with L=limsup |a_n|^(1/n), allowing zero terms: absolute convergence for L<1 and divergence for L>1; L=1 is inconclusive. An ordinary root limit is a special case.
  • Radius and endpoints of a power series ▶
    State the radius theorem for ∑a_n(x−c)^n: absolute convergence for |x−c|<R and divergence for |x−c|>R; test each real endpoint separately. Include R=0 and R=∞, and use the root/ratio test when applicable.
  • The p-series threshold ▶
    Use the integral test for p>0 to show Σn^(−p) converges exactly when p>1. For p≤0 use the necessary term test instead of applying a decreasing-function hypothesis incorrectly.
SUBTOPIC 10 · 6 concepts

Absolute convergence and rearrangements

Absolute and conditional convergence, including complex terms and the contrasting behavior of rearrangements.

  • Series with complex terms ▶
    Define convergence of a complex series through real and imaginary partial sums, equivalently the modulus of their error tending to zero. Test absolute convergence using ∑|z_n| rather than the signed component sums.
  • Absolute convergence implies convergence ▶
    Prove absolute convergence implies convergence by applying the Cauchy criterion to tails and bounding |∑a_k| by ∑|a_k|. The converse need not hold. For complex series, apply the real result componentwise using |Re z|, |Im z|≤|z|.
  • Classifying conditional convergence ▶
    Classify a convergent real series as absolutely or conditionally convergent by separately testing ∑|a_n|. A divergent series is neither category.
  • Rearranging an absolutely convergent series ▶
    Prove that every permutation of an absolutely convergent real or complex series converges to the same sum by controlling the omitted modulus tail. A rearrangement is a bijection of term indices, with no omissions or duplications.
  • Positive and negative parts of a conditional series ▶
    For a conditionally convergent real series define a_n⁺=max(a_n,0), a_n⁻=max(−a_n,0). Prove ∑a_n⁺=∑a_n⁻=+∞ from a_n=a_n⁺−a_n⁻ and |a_n|=a_n⁺+a_n⁻.
  • Rearranging a conditionally convergent series ▶
    Construct a rearrangement of a conditionally convergent real series to any prescribed real target by alternating positive and negative blocks that cross it. Vanishing terms force overshoots to shrink; every original index, including zeros, must eventually appear. Take each positive block until the sum strictly exceeds the target, then each negative block until it is strictly below it, even if an intermediate sum equals the target.
SUBTOPIC 11 · 4 concepts

Refined tests for positive series

Kummer’s comparison principle and the Raabe and Gauss tests it yields, for positive series where the ratio test is inconclusive, followed by a strategy for choosing a test.

  • Kummer’s test ▶
    For aₙ>0 and positive cₙ set Kₙ=cₙaₙ/aₙ₊₁−cₙ₊₁. Prove: if Kₙ≥δ>0 eventually then ∑aₙ converges; if ∑1/cₙ diverges and Kₙ≤0 eventually then ∑aₙ diverges. Convergence telescopes cₙaₙ−cₙ₊₁aₙ₊₁≥δaₙ₊₁; divergence compares aₙ with a multiple of 1/cₙ.
  • Raabe’s test ▶
    For aₙ>0 with R=lim n(aₙ/aₙ₊₁−1): the series converges if R>1 and diverges if R<1; R=1 is inconclusive. Derive both halves from Kummer’s test with cₙ=n, and use the test where the ratio limit equals 1.
  • Gauss’s test ▶
    If aₙ>0 and aₙ/aₙ₊₁=1+h/n+bₙ/nʳ with r>1 and (bₙ) bounded, then ∑aₙ converges when h>1 and diverges when h≤1. The cases h≠1 follow from Raabe’s test; the boundary case h=1 uses Kummer with cₙ=n log n (proof idea only). Expand the ratio in powers of 1/n to read off h.
  • Choosing a test for a positive series ▶
    Select a test from the form of aₙ: factorials or products of linear factors suggest the ratio test, then Raabe or Gauss if the ratio limit is 1; nth powers suggest the root test; rational or root expressions suggest comparison with a p-series. Verify the chosen test’s hypotheses and refine only when a coarser test is inconclusive.
SUBTOPIC 12 · 11 concepts

Limits and continuity

Function limits, relative continuity, algebra, the intermediate value theorem, uniform continuity and continuous inverses.

  • Accumulation points and punctured neighborhoods ▶
    Define an accumulation point a of D by every punctured neighborhood of a meeting D; a need not belong to D. Distinguish an isolated point from an accumulation point.
  • Limits of real functions ▶
    Define lim_{x→a,x∈D}f(x)=L at an accumulation point a of D using 0<|x−a|<δ. The function need not be defined at a; the limit ignores its value there. Distinguish one-sided limits and require their agreement when both sides accumulate.
  • The sequential criterion for a function limit ▶
    Prove the sequential criterion for a function limit at an accumulation point: all domain sequences approaching a, with no term equal to a, yield function values approaching L. One favorable path does not suffice.
  • Continuity at a point ▶
    Define continuity of f:D→R at a∈D using relative neighborhoods, including endpoints and isolated domain points. At an accumulation point, continuity means the limit equals f(a); existence of a limit alone is insufficient.
  • Function limits as the input tends to infinity ▶
    Define f(x)→L as x→+∞ using ∀ε>0 ∃M ∀x in the domain with x>M, |f(x)−L|<ε, for a domain unbounded above. State the analogous negative-infinity and infinite-output definitions; infinity is not a domain point.
  • Algebra of function limits ▶
    Use the sequential criterion to transfer uniqueness, sum, product, quotient and squeeze laws from sequences to function limits at accumulation points. A quotient law requires a nonzero limiting denominator; continuous arithmetic combinations inherit these rules.
  • Composition of continuous functions ▶
    Prove composition preserves continuity when f is continuous at a and g is continuous at f(a), with f(D) in g’s domain. The inner tolerance must be chosen to meet the outer function’s requirement.
  • The intermediate value theorem ▶
    Prove a continuous function on an interval takes all intermediate values between its values at any two points, using completeness and continuity. The conclusion guarantees existence, not uniqueness or monotonicity.
  • Uniform continuity ▶
    Distinguish ∀ε∃δ∀x,y∈D from pointwise continuity’s point-dependent δ. Prove uniform continuity by a global estimate and refute it using arbitrarily close pairs with a fixed positive output gap.
  • Continuity and open preimages ▶
    Prove f:D→R is continuous everywhere exactly when the preimage of every open subset of R is relatively open in D. Preimage notation does not assume injectivity or an inverse function.
  • The inverse of a continuous monotone function ▶
    For a continuous strictly monotone function on an interval, prove its inverse on its image is continuous, using order and relative neighborhoods at possible endpoints. Mere continuity does not guarantee an inverse.
SUBTOPIC 13 · 8 concepts

Real-line topology and compactness

Open and closed sets, compactness and its consequences for continuous functions on the real line.

  • Open sets on the real line ▶
    Define an open subset of R by a neighborhood around each of its points. Prove arbitrary unions and finite intersections are open; an infinite intersection of open sets need not be open. Both ∅ and R are open.
  • Relative openness ▶
    Characterize U⊆D as relatively open when U=D∩V for some open V⊆R, equivalently each point has a neighborhood within D contained in U. Relative openness depends on D.
  • Closed sets on the real line ▶
    Define closed subsets of R by open complements and prove the equivalent criterion that every convergent sequence of set members has its limit in the set. Closed is not the negation of open: sets can be both or neither.
  • Open covers and finite subcovers ▶
    Define an open cover and a finite subcover; the chosen finite sets must come from the given cover. Prove failure of a finite subcover in a concrete example by considering the largest selected index.
  • Compact subsets of the real line ▶
    Define compactness by finite subcovers and use the Heine–Borel theorem in R: compact exactly when closed and bounded. State the theorem explicitly rather than treating boundedness alone as sufficient.
  • Continuity on a compact set is uniform ▶
    Prove continuity on a compact domain implies uniform continuity, using finitely many local neighborhoods and a single positive minimum radius. Compactness is sufficient, not necessary, for a particular function to be uniformly continuous.
  • Continuous images of compact sets ▶
    Prove a continuous image of a compact set is compact by pulling an arbitrary image cover back to the domain and then pushing the finite selection forward. Neither injectivity nor surjectivity onto R is needed.
  • The extreme value theorem ▶
    Prove a continuous real function on a nonempty compact set attains a finite maximum and minimum: its image is compact, hence bounded and closed, and its sup/inf belong to it.
SUBTOPIC 14 · 12 concepts

Differentiation

Derivatives as justified limits, core differentiation proofs, mean value theorems and controlled approximation.

  • The derivative as a limit ▶
    Define a finite derivative by the difference-quotient limit at an interior point. Interpret it as the coefficient of a first-order linear approximation; unequal one-sided quotient limits rule out differentiability.
  • Differentiability implies continuity ▶
    Prove differentiability at an interior point implies continuity by factoring the increment into h times its difference quotient. Continuity alone does not imply differentiability.
  • A derivative at an interior extremum ▶
    Prove a differentiable function at an interior local extremum has zero derivative by opposite signs of one-sided quotients. A stationary point need not be an extremum, and boundary extrema need not be stationary.
  • The chain rule ▶
    Prove (f∘g)′(a)=f′(g(a))g′(a) from first-order error formulas when g is differentiable at a and f at g(a). Avoid dividing by g(a+h)−g(a), which can be zero.
  • Deriving the product rule ▶
    Prove (fg)′=f′g+fg′ at a point where both factors are differentiable, by adding and subtracting a cross term and using their continuity. The derivative is not f′g′.
  • Rolle’s theorem ▶
    Prove Rolle’s theorem with continuity on [a,b], differentiability on (a,b), a<b and equal endpoint values. Combine attainment of extrema with Fermat’s theorem; handle the constant case separately.
  • The mean value theorem ▶
    Prove the mean value theorem under continuity on [a,b] and differentiability on (a,b), a<b, by subtracting the secant line and applying Rolle. The theorem asserts some interior point, not a prescribed one.
  • Cauchy’s generalized mean value theorem ▶
    Prove Cauchy’s mean value identity [f(b)−f(a)]g′(c)=[g(b)−g(a)]f′(c) under the usual continuity/differentiability assumptions. Forming a ratio requires the relevant denominators to be nonzero.
  • Taylor approximation with a remainder bound ▶
    State Taylor’s theorem with Lagrange remainder for f having n+1 continuous derivatives on an interval containing a and x; bound the remainder by M|x−a|^(n+1)/(n+1)!. Finite-order approximation is not automatically an infinite Taylor-series representation.
  • What a derivative sign implies on an interval ▶
    Deduce from the mean value theorem that f′≥0 implies f is nondecreasing and f′>0 implies f is strictly increasing on an interval. Derivative zero everywhere implies a constant function; statements concern an interval, not disconnected pieces.
  • L’Hôpital’s rule for a 0/0 limit ▶
    State the one-sided finite-point 0/0 rule: differentiable f,g on a punctured interval, g′ nonzero there, both f and g tending to zero, and f′/g′ tending to L imply f/g tends to L. Treat L as finite or extended real; verify both sides for a two-sided limit.
  • Derivatives have the intermediate value property ▶
    Prove Darboux’s theorem: if f is differentiable on [a,b], with one-sided derivatives f′(a) and f′(b) at the endpoints, and y lies strictly between f′(a) and f′(b), then f′(c)=y for some c in (a,b), by applying the extreme value theorem and Fermat’s theorem to g(x)=f(x)−yx. Consequently a derivative has no jump discontinuity, although it need not be continuous.
SUBTOPIC 15 · 4 concepts

Indeterminate forms and one-variable extrema

L’Hôpital’s rule beyond the finite-point 0/0 case, reduction of every indeterminate form, derivative tests for local extrema and absolute extrema on intervals.

  • L’Hôpital’s rule for ∞/∞ and limits at infinity ▶
    State the ∞/∞ form: f,g differentiable on a punctured one-sided interval, g′≠0 there, |g|→∞ and f′/g′→L imply f/g→L; f need not tend to infinity. Extend the 0/0 and ∞/∞ forms to x→±∞ by substituting x=1/t, and re-check the hypotheses before each repeated application.
  • Rewriting 0·∞, ∞−∞ and exponent forms ▶
    Reduce 0·∞ to a quotient via fg=f/(1/g), ∞−∞ to a single quotient by combining or factoring, and 1^∞, 0⁰, ∞⁰ by logarithms: if f>0 and g·log f→M then f^g→e^M by continuity of exp. Recognise forms that are not indeterminate, such as 0^∞ and ∞·∞.
  • Derivative tests for local extrema ▶
    If f is continuous at an interior point c and differentiable near c (except possibly at c), a change of f′ from positive to negative across c gives a strict local maximum (negative to positive, a minimum). If f′(c)=⋯=f⁽ⁿ⁻¹⁾(c)=0≠f⁽ⁿ⁾(c) with f⁽ⁿ⁾ continuous near c, then n even gives a strict extremum whose type is the sign of f⁽ⁿ⁾(c), and n odd gives none. Prove the higher-order test from Taylor’s theorem.
  • Absolute extrema on a closed interval ▶
    For f continuous on [a,b] the absolute maximum and minimum exist and occur at endpoints, at interior points with f′=0, or at interior points where f is not differentiable; compare f at every candidate. On an open or unbounded interval, argue existence separately, for example from the limits at the ends.
SUBTOPIC 16 · 16 concepts

Riemann integration

Darboux construction, integrability criteria and rigorous integration rules.

  • Partitions and upper and lower sums ▶
    Define a finite ordered partition of [a,b], a<b, and Darboux sums using infima/suprema of a bounded function on each closed subinterval. Refinement cannot decrease lower sums or increase upper sums; extrema need not be attained.
  • Riemann integrability ▶
    Define lower and upper integrals as suprema of lower sums and infima of upper sums for bounded f on [a,b]. Riemann integrability means these coincide; a function can be bounded but nonintegrable.
  • The upper–lower sum criterion ▶
    Prove the Darboux criterion: bounded f on [a,b] is integrable iff every ε>0 admits a partition with U(f,P)−L(f,P)<ε. Common refinement permits combining separate upper and lower approximations.
  • Why bounded monotone functions are integrable ▶
    Prove a bounded monotone function on a compact interval is Riemann integrable using equal-width partitions and a telescoping oscillation sum. Continuity is not required.
  • Linearity of the integral ▶
    Prove integrability and linearity for real linear combinations of integrable functions, using oscillation estimates for integrability and upper/lower bounds for the integral identity. Negative scalars reverse infimum/supremum roles.
  • Additivity over adjacent intervals ▶
    Prove restriction to adjacent subintervals preserves Riemann integrability and that ∫_a^b f=∫_a^c f+∫_c^b f. State also the converse gluing of two integrable pieces and the sign convention for reversed limits.
  • Tagged Riemann sums and mesh size ▶
    For bounded f on [a,b], state the equivalence between Darboux integrability and convergence of every tagged Riemann sum as partition mesh tends to zero. Tags lie in their own subintervals; one selected sequence of sums is insufficient in general.
  • Bounding an integral by a uniform bound ▶
    For integrable functions derive order bounds and |∫f|≤∫|f|≤M(b−a) when |f|≤M, explaining integrability of |f| via its oscillation bound. The interval length and boundedness assumptions are explicit.
  • Differentiating an accumulation function ▶
    For Riemann-integrable f on [a,b], prove F(t)=∫_a^t f is differentiable at every interior point x where f is continuous, with F′(x)=f(x). Control both positive and negative increments. First use boundedness of f to show F is Lipschitz, hence continuous on the entire closed interval, including its endpoints.
  • Why continuous functions are integrable ▶
    Prove every continuous real function on [a,b] is Riemann integrable: compactness supplies boundedness and uniform continuity, and a sufficiently fine partition controls the Darboux gap.
  • Evaluating an integral with an antiderivative ▶
    Prove ∫_a^b f=G(b)−G(a) when f is continuous on [a,b], G is continuous there and G′=f inside, by comparing G with the accumulation integral and applying the mean value theorem.
  • Why integration by parts works ▶
    Derive ∫_a^b F G′=[FG]_a^b−∫_a^b F′G for continuously differentiable F,G on [a,b], using the product rule and fundamental theorem. Both boundary terms and the minus sign are essential.
  • Why change of variables works ▶
    For continuous f on an interval containing g([a,b]) and continuously differentiable g, derive ∫_a^b f(g(t))g′(t)dt=∫_{g(a)}^{g(b)}f(u)du. This formula does not require g to be one-to-one; oriented limits account for reversal.
  • Products and absolute values of integrable functions ▶
    Prove that |f| is integrable when f is, since ||f(x)|−|f(y)||≤|f(x)−f(y)| bounds its oscillation on each subinterval. For f and g integrable on [a,b], deduce that f² and fg=((f+g)²−(f−g)²)/4 are integrable, and that 1/g is integrable when also |g|≥c>0 on [a,b]. Conclude |∫f|≤∫|f|. Integrability of |f| does not imply integrability of f.
  • The first mean value theorem for integrals ▶
    Prove: if f is continuous on [a,b] and g is integrable with g≥0, then ∫fg=f(c)∫g for some c in [a,b]; with g=1, ∫f=f(c)(b−a). Use the extreme and intermediate value theorems on m∫g≤∫fg≤M∫g. The sign condition on g cannot be dropped.
  • The second mean value theorem for integrals ▶
    State it: if f is integrable and g is monotone on [a,b], then ∫fg=g(a)∫_a^ξ f+g(b)∫_ξ^b f for some ξ in [a,b] (Bonnet’s form: ∫fg=g(a)∫_a^ξ f when g is nonnegative and decreasing). Explain the proof for continuous f and continuously differentiable monotone g via integration by parts with F(x)=∫_a^x f and the first mean value theorem. Use it to bound oscillatory integrals.
SUBTOPIC 17 · 6 concepts

Improper integrals

Improper integrals as independently convergent endpoint limits, with comparison and integral tests.

  • An integral with an infinite endpoint ▶
    Define an integral over an infinite interval as a limit of proper integrals, assuming local Riemann integrability. For the whole real line both one-sided integrals must converge separately; a symmetric principal value is a different notion.
  • An integral with a singular endpoint ▶
    Define an improper integral at a finite singular endpoint by truncation. An interior singularity requires separate convergence on its two sides; finite endpoint values do not repair unbounded behavior nearby.
  • The integral test for a positive series ▶
    Prove the integral test for nonnegative nonincreasing locally Riemann-integrable f on [1,∞): compare sums with integrals over unit intervals. Derive matching convergence and bounds on the remainder; finitely many initial terms do not affect convergence.
  • Comparison tests for improper integrals ▶
    Apply direct comparison for nonnegative functions and limit comparison when their ratio tends to a finite positive number at the problematic endpoint. Check local integrability away from that endpoint before concluding.
  • Power integrals at zero and infinity ▶
    Compute truncated integrals to prove ∫_1^∞ x^(−p) converges exactly for p>1 and ∫_0^1 x^(−p) exactly for p<1. Treat p=1 using logarithms.
  • Absolute and conditional integral convergence ▶
    Define absolute and conditional convergence of signed improper integrals and explain why absolute convergence implies convergence. Use truncated-tail estimates; cancellation can give conditional convergence without absolute convergence. For the implication, split f into its nonnegative positive and negative parts, each bounded by |f|, and apply improper comparison and linearity.
SUBTOPIC 18 · 16 concepts

Sequences and series of functions

Pointwise versus uniform convergence, justified limit interchanges and polynomial approximation.

  • Pointwise convergence ▶
    Define pointwise convergence on a specified domain with a separate N(ε,x) for each fixed x. Determine the limit of x^n on [0,1], including both endpoints; the limiting function can be discontinuous.
  • Uniform convergence ▶
    Define uniform convergence with one N(ε) working for every x in the domain, and show it implies pointwise convergence. Compare x/n on [0,1] with x^n on [0,1]; changing the domain can change uniformity.
  • The supremum error criterion ▶
    Characterize uniform convergence by sup over the domain of |f_n−f| tending to zero. The supremum need not be attained and may be infinite; a single fixed x does not control the supremum.
  • A series of functions via its partial sums ▶
    Treat Σg_n as a sequence of partial-sum functions, with pointwise and uniform convergence defined for those sums. Analyze the geometric function series on [0,1/2] and [0,1).
  • The uniform Cauchy criterion ▶
    Prove real-valued functions converge uniformly exactly when their differences satisfy the Cauchy condition with one index threshold valid for all domain points. Use completeness pointwise, then pass to the limit in a uniform tail bound.
  • Continuity of a uniform limit ▶
    Prove a uniform limit of continuous functions is continuous at each domain point, using three error terms and fixing one approximant before choosing δ. Continuity is relative to the specified domain.
  • Integrability of a uniform limit ▶
    Prove a uniform limit of Riemann-integrable functions on a fixed compact interval is Riemann integrable. Establish boundedness of the limit and estimate its Darboux gap using one approximant and its partition.
  • The Weierstrass approximation theorem ▶
    State the Weierstrass approximation theorem: every continuous real function on a compact interval admits polynomial approximations with arbitrarily small uniform error. Apply the theorem without claiming its proof has been taught.
  • Integrating a uniform limit ▶
    Prove integration commutes with a uniform limit of Riemann-integrable functions on [a,b], using the error bound (b−a)sup|f_n−f| and integrability of the limit.
  • Differentiating a limit of functions ▶
    State and apply this sufficient differentiation theorem: f_n are C¹ on [a,b], f_n′ converge uniformly there, and f_n(x0) converge at one point. Then f_n converge uniformly to a C¹ function whose derivative is the derivative limit, with endpoint derivatives interpreted one-sided.
  • The Weierstrass M-test ▶
    Apply the Weierstrass M-test: if |g_n(x)|≤M_n for every x and ΣM_n converges, then Σg_n converges uniformly and absolutely. The bounds must be independent of x; the test is sufficient, not necessary.
  • Term-by-term integration ▶
    Integrate a uniformly convergent series of Riemann-integrable functions term by term on a compact interval by applying the uniform integral theorem to partial sums. State the interval and convergence justification before the interchange.
  • Term-by-term differentiation ▶
    For C¹ terms on [a,b], uniform convergence of the derivative series and convergence of the original series at one point justify termwise differentiation of the sum. Apply the sequence theorem to partial sums.
  • Uniform convergence inside a power-series radius ▶
    Prove a power series converges uniformly on every closed interval strictly inside its radius of convergence. Choose an intermediate radius and dominate the terms by a convergent geometric series.
  • A continuous highly oscillating series ▶
    Prove uniform convergence and continuity on R of W(x)=Σ_{k≥0}4^(−k)cos(160^k x), using the geometric majorant 4^(−k) and continuity of each partial sum.
  • Continuous everywhere, differentiable nowhere ▶
    State the result established for this exact function in the cited MIT lecture that this Weierstrass function is nowhere differentiable; its full proof is deferred. Explain that continuity does not imply differentiability at even one point, and divergence of the formal derivative series alone is not a proof of nondifferentiability.
SUBTOPIC 19 · 11 concepts

Partial derivatives and extrema

Limits, linear approximation and extrema in two and three variables.

  • Limits in two and three dimensions ▶
    Define limits and continuity in R² and R³ using Euclidean distance. A norm estimate proves a limit along every approach; different path limits disprove existence, while agreement along finitely many paths cannot prove it.
  • A partial derivative ▶
    Define each partial derivative by varying one coordinate and fixing the others. Compute partials at a specified point using the appropriate one-variable difference quotient or differentiation rule.
  • Partial derivatives do not guarantee continuity ▶
    Use f(x,y)=xy/(x²+y²) away from zero, with f(0,0)=0, to show both partial derivatives can exist at a point where the function is discontinuous. Axis information does not control diagonal approaches.
  • Total differentiability ▶
    Define differentiability by a linear approximation whose error divided by the increment norm tends to zero. For a scalar function the linear map is the gradient dot the increment; differentiability implies continuity and existence of partial derivatives.
  • The Hessian matrix ▶
    Form the Hessian of second partial derivatives in R² or R³. Continuous mixed second partials in a neighborhood guarantee symmetry; existence alone does not justify exchanging derivative order.
  • Compact domains in Euclidean space ▶
    Use the finite-dimensional Heine–Borel theorem and extreme value theorem: a continuous real function on a nonempty closed bounded subset of R² or R³ attains its extrema. Closedness and boundedness both matter.
  • The gradient and stationary points ▶
    Apply the necessary stationary-point condition: at an interior local extremum of a differentiable scalar function, its gradient is zero. A zero gradient alone does not distinguish minima, maxima and saddles.
  • Continuous partials guarantee differentiability ▶
    State the sufficient criterion: first partial derivatives exist in a neighborhood and are continuous at the point, then the function is differentiable there. Apply it to polynomial functions; continuity of partials is sufficient, not necessary.
  • Directional derivatives of a differentiable function ▶
    Define a directional derivative along a unit vector and derive D_v f=∇f·v when f is differentiable. Existence of directional derivatives alone does not establish total differentiability.
  • Classifying a stationary point ▶
    State the second-derivative test at a stationary point of a C² function: positive definite Hessian gives a strict minimum, negative definite a strict maximum, and indefinite a saddle. A singular semidefinite Hessian is inconclusive; give the two-variable determinant form.
  • Global extrema on a compact domain ▶
    Find absolute extrema on a compact region by comparing all interior stationary candidates and boundary extrema, including corners and endpoints. Parameterizing the boundary turns its restriction into a lower-dimensional problem.
SUBTOPIC 20 · 9 concepts

Two-variable calculus and constrained extrema

Repeated and double limits, equality of mixed partials, the chain rule and Taylor’s theorem in two variables, then the implicit function theorem, Jacobians and Lagrange’s method with one or two constraints.

  • Repeated limits and the double limit ▶
    Distinguish the double limit of f(x,y) as (x,y)→(a,b) from the repeated limits lim_x lim_y f and lim_y lim_x f. If the double limit L exists and each inner limit exists, the repeated limit equals L; repeated limits can exist and differ, or agree while the double limit fails to exist.
  • When mixed partial derivatives agree ▶
    State Schwarz’s theorem: if f_x, f_y and (f_x)_y exist near a point and (f_x)_y is continuous there, then (f_y)_x exists there and equals (f_x)_y; so C² functions have equal mixed partials. Young’s theorem reaches the same conclusion when f_x and f_y are differentiable at the point. Explain the proof idea: the second difference f(a+h,b+k)−f(a+h,b)−f(a,b+k)+f(a,b), divided by hk, evaluated by the mean value theorem.
  • The chain rule in several variables ▶
    If x(t), y(t) are differentiable at t and f is differentiable at (x(t),y(t)), then (f∘(x,y))′=f_x·x′+f_y·y′; with x(u,v), y(u,v) the same rule gives f_u and f_v. Differentiability of f, not mere existence of partials, is required. Use the rule to change variables, for example to polar coordinates.
  • Taylor’s theorem in two variables ▶
    For f with continuous partials of order n+1 on an open convex set containing (a,b) and (a+h,b+k), apply the one-variable theorem to φ(t)=f(a+th,b+tk): f(a+h,b+k) is the sum of (h∂_x+k∂_y)ᵐf(a,b)/m! for m≤n plus a remainder at an intermediate point. Write the second-order expansion with the Hessian quadratic form.
  • The implicit function theorem in two variables ▶
    State the theorem: if F has continuous partials near (a,b), F(a,b)=0 and F_y(a,b)≠0, then near a there is a unique continuous y=φ(x) with φ(a)=b and F(x,φ(x))=0, and φ′=−F_x/F_y. Explain the existence argument: F_y keeps one sign near (a,b), so F(x,·) is strictly monotone and the intermediate value theorem gives exactly one root.
  • Jacobians ▶
    Define the Jacobian ∂(u,v)/∂(x,y) as the determinant of the matrix of first partials. For differentiable transformations use the chain property ∂(u,v)/∂(r,s)=∂(u,v)/∂(x,y)·∂(x,y)/∂(r,s); for mutually inverse differentiable transformations the Jacobians at corresponding points multiply to 1. A C¹ map with nonzero Jacobian at a point has a C¹ local inverse there (stated; proof deferred); nonzero Jacobian is sufficient, not necessary, for mere local invertibility.
  • Derivatives of implicit functions from two equations ▶
    If F,G are C¹ near a point satisfying F=G=0 with ∂(F,G)/∂(u,v)≠0 there, the equations define unique C¹ functions u(x,y), v(x,y) nearby (stated); differentiate both equations and solve by Cramer’s rule, e.g. u_x=−[∂(F,G)/∂(x,v)]/[∂(F,G)/∂(u,v)]. For F(x,y)=0 with F C² and F_y≠0, differentiate φ′=−F_x/F_y again along y=φ(x) to get φ″.
  • Lagrange’s method of multipliers ▶
    To extremize f(x,y,z) subject to g=0 with f,g C¹ and ∇g≠0 on the constraint, every local constrained extremum satisfies ∇f=λ∇g and g=0. The condition is necessary, not sufficient: decide maxima and minima by comparing candidate values on a compact constraint set or by a separate existence argument. Where ∇f≠0, interpret it as tangency of the level surface of f to the constraint surface.
  • Constrained extrema with two constraints ▶
    Let f,g,h be C¹ near the constraint set g=h=0. If ∇g and ∇h are linearly independent at a local constrained extremum, then ∇f=λ∇g+μ∇h there. Classify candidates by comparing values on a compact constraint curve.