Math 2.0

Counterexamples

An object that shows that a statement is false.

Ideas 1–71 of 71.

NameKindSubject
Non-uniqueness for a non-Lipschitz equationCounterexampleOrdinary differential equations
Weierstrass functionCounterexampleReal functions
Function with unequal mixed partial derivativesCounterexampleReal functions
The Dirichlet function is not Riemann integrableCounterexampleReal functions
Z[x] is not a principal ideal domainCounterexampleCommutative algebra
Smooth non-analytic functionCounterexampleReal functions
x^2 is not uniformly continuous on the real lineCounterexampleReal functions
sin(1/x) has no limit at 0CounterexampleReal functions
Pairwise independent events that are not mutually independentCounterexampleProbability theory and stochastic processes
Z[√-5] is not a UFDCounterexampleNumber theory
Separately continuous but not continuous functionCounterexampleReal functions
Uncorrelated random variables that are not independentCounterexampleProbability theory and stochastic processes
Irreducible elements need not be primeCounterexampleCommutative algebra
Absolute value is continuous but not differentiable at 0CounterexampleReal functions
Function for which the iterated integrals differCounterexampleReal functions
Curl-free field that is not conservativeCounterexampleReal functions
Topologist's sine curveCounterexampleGeneral topology
Non-commutativity of matrix multiplicationCounterexampleLinear and multilinear algebra; matrix theory
Converse of Lagrange's theorem fails for the alternating group A4CounterexampleGroup theory and generalizations
A continuous nowhere differentiable functionCounterexampleReal functions
Dirichlet functionCounterexampleReal functions
Partial derivatives do not imply differentiabilityCounterexampleReal functions
x^2 sin(1/x) is differentiable with discontinuous derivativeCounterexampleReal functions
x^n converges pointwise but not uniformly on [0, 1]CounterexampleSequences, series, summability
The complex conjugate is not holomorphicCounterexampleFunctions of a complex variable
Function with no Laplace transformCounterexampleIntegral transforms, operational calculus
Long lineCounterexampleGeneral topology
Rossi's non-embeddable CR structureCounterexampleSeveral complex variables and analytic spaces
Kolmogorov's everywhere divergent Fourier seriesCounterexampleHarmonic analysis on Euclidean spaces
Hodges' superefficient estimatorCounterexampleStatistics
Knaster–Kuratowski fanCounterexampleGeneral topology
Sorgenfrey planeCounterexampleGeneral topology
Weierstrass's counterexample to the Dirichlet principleCounterexampleCalculus of variations and optimal control; optimization
Lewy's exampleCounterexamplePartial differential equations
Zaremba's punctured ballCounterexamplePotential theory
Failure of Kalman's conjectureCounterexampleSystems theory; control
Telescope conjectureCounterexampleAlgebraic topology
Lindley's paradoxCounterexampleStatistics
Fort spaceCounterexampleGeneral topology
Stein's exampleCounterexampleStatistics
Counterexample to the Markus–Yamabe conjectureCounterexampleOrdinary differential equations
Amitsur's noncrossed productCounterexampleAssociative rings and algebras
Conway base 13 functionCounterexampleReal functions
Hausdorff paradoxCounterexampleMeasure and integration
Borel–Kolmogorov paradoxCounterexampleProbability theory and stochastic processes
Comb spaceCounterexampleGeneral topology
Failure of Aizerman's conjectureCounterexampleSystems theory; control
Burali-Forti paradoxCounterexampleMathematical logic and foundations
Hartogs triangleCounterexampleSeveral complex variables and analytic spaces
Tychonoff plankCounterexampleGeneral topology
Line with two originsCounterexampleGeneral topology
Wente torusCounterexampleDifferential geometry
Lebesgue spineCounterexamplePotential theory
Arens–Fort spaceCounterexampleGeneral topology
Continuous function with divergent Fourier seriesCounterexampleHarmonic analysis on Euclidean spaces
Cantor functionCounterexampleReal functions
Moments of the log-normal distributionCounterexampleIntegral transforms, operational calculus
Hedetniemi's conjectureCounterexampleCombinatorics
Alexander horned sphereCounterexampleManifolds and cell complexes
Wild knotCounterexampleManifolds and cell complexes
Volterra's functionCounterexampleReal functions
Tychonoff's non-uniqueness exampleCounterexamplePartial differential equations
Moore planeCounterexampleGeneral topology
Hadamard's example of ill-posednessCounterexamplePartial differential equations
Simpson's paradoxCounterexampleStatistics
Borsuk's conjectureCounterexampleConvex and discrete geometry
Vitali setCounterexampleMeasure and integration
Sorgenfrey lineCounterexampleGeneral topology
Nagata's example of a Noetherian ring of infinite dimensionCounterexampleCommutative algebra
Hirsch conjectureCounterexampleConvex and discrete geometry
Enflo's counterexample to the approximation problemCounterexampleFunctional analysis