Analysis
191 questions across 7 topics. Also examined in CNET, ISS, GATE ST, GATE DA, JAM MS.
Sequences and Series
- 4Series — divergence by comparisonMSET June 2025Sequences and SeriesMedium
- 5Sequences — when the liminf/limsup chain is not strictMSET June 2025Sequences and SeriesHard
- 6Uniform convergence on [0,1] versus [0,tfrac12]MSET June 2025Sequences and SeriesHard
- 6Set sequences — liminf/limsup of shrinking irrational setsMSET April 2024Sequences and SeriesMedium
- 8Cardinality — sequences over a four-letter alphabetMSET April 2024Sequences and SeriesHard
- 10Double limits — the standard axes counterexampleMSET April 2024Sequences and SeriesHard
- 22Complex logarithm — two familiar-looking identities that breakMSET April 2024Sequences and SeriesHard
- 26Recovering f from its values on a convergent sequenceMSET April 2024Sequences and SeriesMedium
- 27Identity theorem — a sequence of zeros needs to accumulateMSET April 2024Sequences and SeriesHard
- 28Liouville-type consequences for f=u+iv entireMSET April 2024Sequences and SeriesHard
- 49Riesz's theorem — compactness of the unit ball characterises finite dimensionMSET April 2024Sequences and SeriesMedium
- 87Coercive functionals on a Banach spaceMSET April 2024Sequences and SeriesMedium
- 6Set limits — every point misses infinitely many termsMSET March 2023Sequences and SeriesMedium
- 10Partial derivatives fail to exist for this classic non-differentiable exampleMSET March 2023Sequences and SeriesMedium
- 29Radius of convergence via the ratio test and eMSET March 2023Sequences and SeriesEasy
- 30Weierstrass's theorem — uniform limits of analytic functions are analyticMSET March 2023Sequences and SeriesEasy
- 31Recovering f from a convergent sequence of values inside a diskMSET March 2023Sequences and SeriesEasy
- 36Finite extensions of mathbf R — only mathbf R and mathbf C existMSET March 2023Sequences and SeriesEasy
- 1Positive convergent sequences — what survives to every subsequenceMSET September 2021Sequences and SeriesHard
- 11Eventually-zero sequences are exactly the countable onesMSET September 2021Sequences and SeriesHard
- 46Consequences of sub-linear growth for a non-constant entire functionMSET September 2021Sequences and SeriesMedium
- 57Finitely-supported subspaces of ell^2 never algebraically sum to all of ell^2MSET September 2021Sequences and SeriesHard
- 1Which limit-inequality statement survives in full generalityMSET December 2020Sequences and SeriesMedium
- 12Every real sequence has a monotone subsequenceMSET December 2020Sequences and SeriesMedium
- 14xsin(1/x) is continuous but nowhere of bounded variation near 0, and not differentiable thereMSET December 2020Sequences and SeriesMedium
- 49The ring of analytic functions is a non-Noetherian integral domainMSET December 2020Sequences and SeriesHard
- 3The positive subsequence (n+1)/n decreases to 1, giving the limsupMSET June 2019Sequences and SeriesEasy
- 14A differentiable function with bounded, oscillating derivative is still of bounded variationMSET June 2019Sequences and SeriesMedium
- 56Stirling's growth versus a root that tames a logarithmMSET June 2019Sequences and SeriesEasy
- 57Uncountably many pm1 sign patterns give unitary diagonal operatorsMSET June 2019Sequences and SeriesMedium
- 86The standard IVP/BVP integral-equation reductions are exactly reversed hereMSET June 2019Sequences and SeriesMedium
- 1The vanishing term 6/n^2 leaves the two limit points pm1MSET January 2018 Paper IISequences and SeriesEasy
- 2Integrating the geometric series sum(-1)^kx^2k gives tan^-1xMSET January 2018 Paper IISequences and SeriesEasy
- 21Heine–Borel in R^2: the parabola is unbounded, the sine curve not closedMSET January 2018 Paper IIISequences and SeriesEasy
- 43an^2=4cdot5^n, so a12=2cdot5^6MSET January 2018 Paper IIISequences and SeriesEasy
- 2sin(npi/3) cycles through pmfracsqrt32 and 0MSET April 2017 Paper IISequences and SeriesEasy
- 23fn keeps a bump of height tfrac12 at x=1/n; gn's bump shrinks like 1/sqrt nMSET April 2017 Paper IISequences and SeriesMedium
Continuity, Uniform Continuity, Differentiability, Mean Value Theorem
- 7Differentiation — evaluating a derivative expressionMSET June 2025Continuity, Uniform Continuity, Differentiability, Mean Value TheoremEasy
- 9Continuity — removing a 0/0 indeterminacyMSET June 2025Continuity, Uniform Continuity, Differentiability, Mean Value TheoremEasy
- 5Real functions — what survives under fmapstofMSET April 2024Continuity, Uniform Continuity, Differentiability, Mean Value TheoremMedium
- 9Uniform continuity — checking three named functionsMSET April 2024Continuity, Uniform Continuity, Differentiability, Mean Value TheoremHard
- 62A textbook Peano non-uniqueness exampleMSET April 2024Continuity, Uniform Continuity, Differentiability, Mean Value TheoremMedium
- 65Lipschitz condition — bounded rectangle versus unbounded stripMSET April 2024Continuity, Uniform Continuity, Differentiability, Mean Value TheoremMedium
- 67An IVP with a knife-edge domain restrictionMSET April 2024Continuity, Uniform Continuity, Differentiability, Mean Value TheoremMedium
- 90Volterra equation — converting to an initial value problemMSET April 2024Continuity, Uniform Continuity, Differentiability, Mean Value TheoremMedium
- 92Solving a Volterra equation by differentiating twiceMSET April 2024Continuity, Uniform Continuity, Differentiability, Mean Value TheoremEasy
- 8Matching value and slope at a piecewise junctionMSET March 2023Continuity, Uniform Continuity, Differentiability, Mean Value TheoremEasy
- 9Filling a removable discontinuity via a half-angle identityMSET March 2023Continuity, Uniform Continuity, Differentiability, Mean Value TheoremEasy
- 61Lipschitz continuity, not mere continuity or boundedness, guarantees uniquenessMSET March 2023Continuity, Uniform Continuity, Differentiability, Mean Value TheoremMedium
- 66Solving a piecewise linear ODE with x forcingMSET March 2023Continuity, Uniform Continuity, Differentiability, Mean Value TheoremMedium
- 92Solving a Volterra equation via differentiationMSET March 2023Continuity, Uniform Continuity, Differentiability, Mean Value TheoremEasy
- 12Bernoulli's inequality — pinning down the valid range of xMSET September 2021Continuity, Uniform Continuity, Differentiability, Mean Value TheoremMedium
- 50Distance between disjoint sets — compactness (not just closedness) is what's neededMSET September 2021Continuity, Uniform Continuity, Differentiability, Mean Value TheoremHard
- 59Transporting continuity across coarser/finer topologiesMSET September 2021Continuity, Uniform Continuity, Differentiability, Mean Value TheoremHard
- 100A clean sufficient condition for absolute continuityMSET September 2021Continuity, Uniform Continuity, Differentiability, Mean Value TheoremMedium
- 2Ruling out three properties of a strictly increasing function via its derivativeMSET December 2020Continuity, Uniform Continuity, Differentiability, Mean Value TheoremEasy
- 31Recognising a semilinear PDE: linear principal part, nonlinear lower-order termMSET December 2020Continuity, Uniform Continuity, Differentiability, Mean Value TheoremEasy
- 42A rank-deficient linear map cannot be an open map into the full codomainMSET December 2020Continuity, Uniform Continuity, Differentiability, Mean Value TheoremHard
- 46A strict derivative bound forces exponential form, and hence f"=cf'MSET December 2020Continuity, Uniform Continuity, Differentiability, Mean Value TheoremMedium
- 58Integration is a bounded operator on C[0,1]; differentiation, under the sup norm, is notMSET December 2020Continuity, Uniform Continuity, Differentiability, Mean Value TheoremHard
- 61Every subfield of R contains Q, hence is dense — so continuity forces the zero functionMSET December 2020Continuity, Uniform Continuity, Differentiability, Mean Value TheoremMedium
- 85Minimizing the distance from a point on y=x^2 to the line x-y=5MSET December 2020Continuity, Uniform Continuity, Differentiability, Mean Value TheoremEasy
- 2sin x-sin ylex-y via the mean value theoremMSET June 2019Continuity, Uniform Continuity, Differentiability, Mean Value TheoremEasy
- 18Cauchy–Riemann equations hold only on the real axis, so f is analytic nowhereMSET June 2019Continuity, Uniform Continuity, Differentiability, Mean Value TheoremEasy
- 35Bounded partial derivatives give Lipschitz; a fractional power in y alone does notMSET June 2019Continuity, Uniform Continuity, Differentiability, Mean Value TheoremMedium
- 41Compactness of [a,b] forces its continuous image to be boundedMSET June 2019Continuity, Uniform Continuity, Differentiability, Mean Value TheoremEasy
- 65A mean-value computation whose printed option is ambiguously typesetMSET June 2019Continuity, Uniform Continuity, Differentiability, Mean Value TheoremHard
- 17e^-1/x glued to 0: smooth at the origin but not analyticMSET January 2018 Paper IIContinuity, Uniform Continuity, Differentiability, Mean Value TheoremMedium
- 22x^3 has derivatives 3xx and 6x, and stops thereMSET January 2018 Paper IIContinuity, Uniform Continuity, Differentiability, Mean Value TheoremEasy
- 23sin x-sin ylex-y: Lipschitz, hence uniformly continuousMSET January 2018 Paper IIContinuity, Uniform Continuity, Differentiability, Mean Value TheoremEasy
- 25log x/x peaks at x=e with value 1/eMSET January 2018 Paper IIContinuity, Uniform Continuity, Differentiability, Mean Value TheoremEasy
- 3f=x on Q, 0 off Q: squeezed to 0 only at the originMSET January 2018 Paper IIIContinuity, Uniform Continuity, Differentiability, Mean Value TheoremEasy
- 36The intermediate value theorem gives a fixed point on [0,1]MSET January 2018 Paper IIIContinuity, Uniform Continuity, Differentiability, Mean Value TheoremEasy
- 1A linear term plus a bounded periodic one is never periodicMSET April 2017 Paper IIContinuity, Uniform Continuity, Differentiability, Mean Value TheoremEasy
- 17xx has derivative 2x, which has a corner at 0MSET April 2017 Paper IIContinuity, Uniform Continuity, Differentiability, Mean Value TheoremEasy
- 21tan x blows up where cos x=0MSET April 2017 Paper IIContinuity, Uniform Continuity, Differentiability, Mean Value TheoremEasy
- 22x^2-y^2=x+yx-y: bounded slope locally, unbounded globallyMSET April 2017 Paper IIContinuity, Uniform Continuity, Differentiability, Mean Value TheoremEasy
- 24The interior critical point gives only a local maximum; the endpoint x=2 gives 2MSET April 2017 Paper IIContinuity, Uniform Continuity, Differentiability, Mean Value TheoremEasy
- 3x^2 on Q, 0 off Q: squeezed only at the originMSET April 2017 Paper IIIContinuity, Uniform Continuity, Differentiability, Mean Value TheoremEasy
- 21xsin(1/x)to0 but sin(1/x) oscillatesMSET April 2017 Paper IIIContinuity, Uniform Continuity, Differentiability, Mean Value TheoremEasy
Metric Spaces, Compactness, Connectedness, Normed Linear Spaces
- 1Metric spaces — connectednessMSET June 2025Metric Spaces, Compactness, Connectedness, Normed Linear SpacesMedium
- 2Metric spaces — the agreement setMSET June 2025Metric Spaces, Compactness, Connectedness, Normed Linear SpacesMedium
- 3Metric spaces — what can never happenMSET June 2025Metric Spaces, Compactness, Connectedness, Normed Linear SpacesHard
- 49Analysis — subsets of ell^infty: every option failsMSET June 2025Metric Spaces, Compactness, Connectedness, Normed Linear SpacesHard
- 51Functional analysis — an evaluation condition defines an open setMSET June 2025Metric Spaces, Compactness, Connectedness, Normed Linear SpacesHard
- 54What every metric space guarantees — and what it doesn'tMSET April 2024Metric Spaces, Compactness, Connectedness, Normed Linear SpacesMedium
- 56Constant functions inside C[0,1] — closed, not dense, not compactMSET April 2024Metric Spaces, Compactness, Connectedness, Normed Linear SpacesHard
- 18Several operators fail to be nilpotent — for overlapping reasonsMSET March 2023Metric Spaces, Compactness, Connectedness, Normed Linear SpacesHard
- 26Open mapping theorem — when the image is forced to be a single pointMSET March 2023Metric Spaces, Compactness, Connectedness, Normed Linear SpacesMedium
- 50Set difference of a half-open rectangle and a disk — when it becomes compactMSET March 2023Metric Spaces, Compactness, Connectedness, Normed Linear SpacesHard
- 52Unions of dense sets are always dense, unlike intersections or differencesMSET March 2023Metric Spaces, Compactness, Connectedness, Normed Linear SpacesMedium
- 57Neither ell^p nor ell^q is a closed subspace of the otherMSET March 2023Metric Spaces, Compactness, Connectedness, Normed Linear SpacesMedium
- 58Both candidate distances fail identity of indiscerniblesMSET March 2023Metric Spaces, Compactness, Connectedness, Normed Linear SpacesMedium
- 91Symmetric kernels give real eigenvalues and orthogonal eigenfunctionsMSET March 2023Metric Spaces, Compactness, Connectedness, Normed Linear SpacesMedium
- 52Continuous surjections preserve densityMSET September 2021Metric Spaces, Compactness, Connectedness, Normed Linear SpacesMedium
- 56A bijective bounded linear map transported to a pair of isometric renormingsMSET September 2021Metric Spaces, Compactness, Connectedness, Normed Linear SpacesHard
- 58A multiplication operator is always normal, but rarely self-adjoint or unitaryMSET September 2021Metric Spaces, Compactness, Connectedness, Normed Linear SpacesHard
- 44The open mapping theorem rules out an image that is merely a circleMSET December 2020Metric Spaces, Compactness, Connectedness, Normed Linear SpacesMedium
- 50(-1,1) is completely metrizable even though its usual metric is not completeMSET December 2020Metric Spaces, Compactness, Connectedness, Normed Linear SpacesHard
- 51An unbounded hyperbola pair versus the compact ell^1-unit circleMSET December 2020Metric Spaces, Compactness, Connectedness, Normed Linear SpacesEasy
- 56The Open Mapping Theorem forces both pullback norms to be equivalent to the originalsMSET December 2020Metric Spaces, Compactness, Connectedness, Normed Linear SpacesHard
- 57A co-isometry that fails every one of normal, self-adjoint, unitary, and T^T=IMSET December 2020Metric Spaces, Compactness, Connectedness, Normed Linear SpacesHard
- 15Compactness, not mere closedness, guarantees positive distance to a disjoint closed setMSET June 2019Metric Spaces, Compactness, Connectedness, Normed Linear SpacesMedium
- 45The open mapping theorem rules out a closed-disc imageMSET June 2019Metric Spaces, Compactness, Connectedness, Normed Linear SpacesEasy
- 50Completeness makes R second category; countability makes Q first categoryMSET June 2019Metric Spaces, Compactness, Connectedness, Normed Linear SpacesEasy
- 52The complement of a nowhere dense set is always denseMSET June 2019Metric Spaces, Compactness, Connectedness, Normed Linear SpacesMedium
- 58Orthogonal halves of C[0,1] that cannot rebuild a function non-zero at 1/2MSET June 2019Metric Spaces, Compactness, Connectedness, Normed Linear SpacesHard
- 21Heine–Borel is special to R^n: closed and bounded need not be compactMSET January 2018 Paper IIMetric Spaces, Compactness, Connectedness, Normed Linear SpacesEasy
- 22Compact iff complete and totally boundedMSET January 2018 Paper IIIMetric Spaces, Compactness, Connectedness, Normed Linear SpacesEasy
- 31A linear span need not be closed in an infinite-dimensional spaceMSET January 2018 Paper IIIMetric Spaces, Compactness, Connectedness, Normed Linear SpacesMedium
- 32Riesz: a compact closed unit ball forces finite dimensionMSET January 2018 Paper IIIMetric Spaces, Compactness, Connectedness, Normed Linear SpacesMedium
- 35A separable dual forces a separable space — not converselyMSET January 2018 Paper IIIMetric Spaces, Compactness, Connectedness, Normed Linear SpacesHard
- 20Adding the limit point 0 makes 1/n compactMSET April 2017 Paper IIIMetric Spaces, Compactness, Connectedness, Normed Linear SpacesEasy
- 31A finite-dimensional space has infinitely many subspaces (over R)MSET April 2017 Paper IIIMetric Spaces, Compactness, Connectedness, Normed Linear SpacesEasy
- 35All four statements are standard theorems — nothing is "not true"MSET April 2017 Paper IIIMetric Spaces, Compactness, Connectedness, Normed Linear SpacesHard
- 75Minkowski's inequalityMSET April 2017 Paper IIIMetric Spaces, Compactness, Connectedness, Normed Linear SpacesEasy
Monotonic Functions, Bounded Variation, Lebesgue Measure & Integral
- 4Integration — Riemann vs. Lebesgue, what discontinuity does not doMSET April 2024Monotonic Functions, Bounded Variation, Lebesgue Measure & IntegralHard
- 59Continuous image of a disk — can't be two disjoint piecesMSET April 2024Monotonic Functions, Bounded Variation, Lebesgue Measure & IntegralMedium
- 105Set theory — the field generated by one setMSET April 2024Monotonic Functions, Bounded Variation, Lebesgue Measure & IntegralHard
- 2A monotone function landing in mathbf Q must be constant somewhereMSET September 2021Monotonic Functions, Bounded Variation, Lebesgue Measure & IntegralHard
- 20A double pole with vanishing residue — none of the printed options surviveMSET September 2021Monotonic Functions, Bounded Variation, Lebesgue Measure & IntegralHard
- 41Integrating an indicator that is 1 almost everywhereMSET September 2021Monotonic Functions, Bounded Variation, Lebesgue Measure & IntegralEasy
- 51Two classical oscillating functions — neither is of bounded variationMSET September 2021Monotonic Functions, Bounded Variation, Lebesgue Measure & IntegralHard
- 98Evaluating a signed measure on an interval straddling the sign changeMSET September 2021Monotonic Functions, Bounded Variation, Lebesgue Measure & IntegralEasy
- 99Absolute continuity of an integral-defined measureMSET September 2021Monotonic Functions, Bounded Variation, Lebesgue Measure & IntegralHard
- 131Set operations on sigma-fields — intersections always survive, unions generally don'tMSET September 2021Monotonic Functions, Bounded Variation, Lebesgue Measure & IntegralHard
- 41An odd integrand over a centrally symmetric slice vanishesMSET December 2020Monotonic Functions, Bounded Variation, Lebesgue Measure & IntegralEasy
- 52Inner regularity forces measure zero, which gives outer-regular open covers of arbitrarily small measureMSET December 2020Monotonic Functions, Bounded Variation, Lebesgue Measure & IntegralMedium
- 98An integral against a sign-changing density is a signed measure, not a measureMSET December 2020Monotonic Functions, Bounded Variation, Lebesgue Measure & IntegralMedium
- 99The intersection of a positive and a negative set is always nu-nullMSET December 2020Monotonic Functions, Bounded Variation, Lebesgue Measure & IntegralEasy
- 131Most uncountable subsets of R fail to be BorelMSET December 2020Monotonic Functions, Bounded Variation, Lebesgue Measure & IntegralMedium
- 13Thomae's function: Riemann integrable, continuous at irrationals, discontinuous at rationalsMSET June 2019Monotonic Functions, Bounded Variation, Lebesgue Measure & IntegralEasy
- 51Countable sets never move Lebesgue measureMSET June 2019Monotonic Functions, Bounded Variation, Lebesgue Measure & IntegralMedium
- 98sin x's integral over an unbounded set is not even well-definedMSET June 2019Monotonic Functions, Bounded Variation, Lebesgue Measure & IntegralMedium
- 99Disjoint supports make two measures mutually singularMSET June 2019Monotonic Functions, Bounded Variation, Lebesgue Measure & IntegralMedium
- 24An increasing function's total variation is f(b)-f(a)MSET January 2018 Paper IIMonotonic Functions, Bounded Variation, Lebesgue Measure & IntegralEasy
- 26Dirichlet's function: rationals and irrationals are both denseMSET January 2018 Paper IIMonotonic Functions, Bounded Variation, Lebesgue Measure & IntegralEasy
- 23Jordan decomposition, and monotone functions are Borel measurableMSET January 2018 Paper IIIMonotonic Functions, Bounded Variation, Lebesgue Measure & IntegralMedium
- 24Dirichlet's function is Lebesgue- but not Riemann-integrable; 1/x is neitherMSET January 2018 Paper IIIMonotonic Functions, Bounded Variation, Lebesgue Measure & IntegralEasy
- 25int0^1x^-1/3dx=tfrac32MSET January 2018 Paper IIIMonotonic Functions, Bounded Variation, Lebesgue Measure & IntegralEasy
- 72Carath\'eodory's criterion: E must split every test set additivelyMSET January 2018 Paper IIIMonotonic Functions, Bounded Variation, Lebesgue Measure & IntegralMedium
- 73Statement I is Halmos's definition of completeness; statement II omits mu(En)<inftyMSET January 2018 Paper IIIMonotonic Functions, Bounded Variation, Lebesgue Measure & IntegralMedium
- 74Sets that differ by a null set have the same integralsMSET January 2018 Paper IIIMonotonic Functions, Bounded Variation, Lebesgue Measure & IntegralEasy
- 75Singularity is closed under sums; nullmu and nuperpmu force nu=0MSET January 2018 Paper IIIMonotonic Functions, Bounded Variation, Lebesgue Measure & IntegralMedium
- 840 on finite sets, infty on infinite ones: finitely but not countably additiveMSET January 2018 Paper IIIMonotonic Functions, Bounded Variation, Lebesgue Measure & IntegralMedium
- 23Borel subsetneq Lebesgue measurableMSET April 2017 Paper IIIMonotonic Functions, Bounded Variation, Lebesgue Measure & IntegralEasy
- 24f measurable does not make f measurableMSET April 2017 Paper IIIMonotonic Functions, Bounded Variation, Lebesgue Measure & IntegralMedium
- 25Lebesgue: functions of bounded variation are differentiable a.e.MSET April 2017 Paper IIIMonotonic Functions, Bounded Variation, Lebesgue Measure & IntegralEasy
- 72mu(A,Delta,B) is a pseudometric: distinct sets can be at distance 0MSET April 2017 Paper IIIMonotonic Functions, Bounded Variation, Lebesgue Measure & IntegralEasy
- 73Fatou's lemma and the monotone convergence theoremMSET April 2017 Paper IIIMonotonic Functions, Bounded Variation, Lebesgue Measure & IntegralEasy
- 74The countable-cover infimum defines the induced outer measureMSET April 2017 Paper IIIMonotonic Functions, Bounded Variation, Lebesgue Measure & IntegralMedium
Elementary Set Theory, Real Number System, Supremum, Infimum
- 11Vector spaces — dimension statementsMSET June 2025Elementary Set Theory, Real Number System, Supremum, InfimumHard
- 54Analysis — a closed dense subset of RMSET June 2025Elementary Set Theory, Real Number System, Supremum, InfimumEasy
- 60Analysis — Cantor's nested interval theoremMSET June 2025Elementary Set Theory, Real Number System, Supremum, InfimumEasy
- 1Sets — evaluating 1+(-1)^nMSET April 2024Elementary Set Theory, Real Number System, Supremum, InfimumEasy
- 7Piecewise functions — computing a preimage setMSET April 2024Elementary Set Theory, Real Number System, Supremum, InfimumMedium
- 1Two disguised descriptions of the same set of odd integersMSET March 2023Elementary Set Theory, Real Number System, Supremum, InfimumEasy
- 5Spotting the pair with genuinely different cardinalitiesMSET March 2023Elementary Set Theory, Real Number System, Supremum, InfimumMedium
- 7The countable-or-co-countable sigma-field — an interval that belongs to neitherMSET March 2023Elementary Set Theory, Real Number System, Supremum, InfimumHard
- 53Convex sets are always connected — and a bowtie union is not even convexMSET March 2023Elementary Set Theory, Real Number System, Supremum, InfimumMedium
- 59Removing a discrete lattice or a parabola from the planeMSET March 2023Elementary Set Theory, Real Number System, Supremum, InfimumHard
- 9Recognising convex sets — a line, a wedge, and an epigraphMSET September 2021Elementary Set Theory, Real Number System, Supremum, InfimumMedium
- 13Counting the crossings of cos x and e^-xMSET September 2021Elementary Set Theory, Real Number System, Supremum, InfimumMedium
- 15Non-degenerate open and connected subsets of mathbf R are always uncountableMSET September 2021Elementary Set Theory, Real Number System, Supremum, InfimumMedium
- 17The zero set of a nonzero entire function is always closed (and discrete, and countable)MSET September 2021Elementary Set Theory, Real Number System, Supremum, InfimumMedium
- 61Finite products preserve both first countability and separabilityMSET September 2021Elementary Set Theory, Real Number System, Supremum, InfimumMedium
- 1Only the closed shrinking intervals share a common pointMSET June 2019Elementary Set Theory, Real Number System, Supremum, InfimumEasy
- 11The supremum of a union is the max of the two supremaMSET June 2019Elementary Set Theory, Real Number System, Supremum, InfimumEasy
- 17e^z= const describes a vertical lineMSET June 2019Elementary Set Theory, Real Number System, Supremum, InfimumEasy
- 61Continuous open surjections push forward a countable baseMSET June 2019Elementary Set Theory, Real Number System, Supremum, InfimumEasy
Functions of Several Variables, Inverse & Implicit Function Theorems
- 8Two variables — partials can exist without continuityMSET June 2025Functions of Several Variables, Inverse & Implicit Function TheoremsHard
- 71The compatibility condition for two first-order PDEsMSET March 2023Functions of Several Variables, Inverse & Implicit Function TheoremsMedium
- 42Implicit function theorem — checking each partial derivative at the originMSET September 2021Functions of Several Variables, Inverse & Implicit Function TheoremsMedium
- 45Maxima and minima of two composed complex functionsMSET September 2021Functions of Several Variables, Inverse & Implicit Function TheoremsHard
- 67Classifying a first-order PDE with z-dependent coefficientsMSET September 2021Functions of Several Variables, Inverse & Implicit Function TheoremsEasy
- 1Along the y-axis f=pm1: no partials, no continuity, no option trueMSET January 2018 Paper IIIFunctions of Several Variables, Inverse & Implicit Function TheoremsMedium
- 2Directional derivative =nabla fcdothat v with hat v=(12,5)/13MSET January 2018 Paper IIIFunctions of Several Variables, Inverse & Implicit Function TheoremsEasy
- 6The Laplacian in polar coordinatesMSET January 2018 Paper IIIFunctions of Several Variables, Inverse & Implicit Function TheoremsEasy
- 7A critical point with a positive definite HessianMSET January 2018 Paper IIIFunctions of Several Variables, Inverse & Implicit Function TheoremsMedium
- 9x^2+y^2+z^2=(x+y+z)^2-2(xy+yz+zx)MSET January 2018 Paper IIIFunctions of Several Variables, Inverse & Implicit Function TheoremsMedium
- 1nabla f=(2xy+7,x^2)=(3,1) at (1,-2)MSET April 2017 Paper IIIFunctions of Several Variables, Inverse & Implicit Function TheoremsEasy
- 2Continuous with zero partials, but the directional derivatives are not linearMSET April 2017 Paper IIIFunctions of Several Variables, Inverse & Implicit Function TheoremsMedium
- 6A critical point with fxx<0 and positive Hessian determinantMSET April 2017 Paper IIIFunctions of Several Variables, Inverse & Implicit Function TheoremsMedium
- 8The Jacobian of f at the originMSET April 2017 Paper IIIFunctions of Several Variables, Inverse & Implicit Function TheoremsEasy
- 9Chain rule: D(fcirc g)(1,2)=Df(0,0),Dg(1,2)MSET April 2017 Paper IIIFunctions of Several Variables, Inverse & Implicit Function TheoremsMedium
Riemann Sums and Riemann Integral, Improper Integrals
- 3A principal-value Mellin-type integralMSET March 2023Riemann Sums and Riemann Integral, Improper IntegralsMedium
- 14Bounded variation implies Riemann integrable — and nothing strongerMSET September 2021Riemann Sums and Riemann Integral, Improper IntegralsHard
- 12Differentiating a self-mirrored integral identity forces fequiv0MSET June 2019Riemann Sums and Riemann Integral, Improper IntegralsEasy
- 100Lebesgue integrability is equivalent to absolute integrabilityMSET June 2019Riemann Sums and Riemann Integral, Improper IntegralsEasy
- 8inta^b(x-a)^-ndx behaves like a p-integral: finite iff n<1MSET January 2018 Paper IIIRiemann Sums and Riemann Integral, Improper IntegralsEasy
- 25f integrable does not make f integrable: pm1 on rationals and irrationalsMSET April 2017 Paper IIRiemann Sums and Riemann Integral, Improper IntegralsEasy