Math 2.0

Counterexamples

An object that shows that a statement is false.

NameTagKindLevelMSC
Non-uniqueness for a non-Lipschitz equation00K7ASZ1PRcounterexampleW034A12
Weierstrass function01A0MK1730counterexampleW026A27
Function with unequal mixed partial derivatives01NN8NY497counterexampleW026B05
The Dirichlet function is not Riemann integrable03Z4JC4YRJcounterexampleW026A42
Z[x] is not a principal ideal domain06E3AG9TM2counterexampleW013F20
Smooth non-analytic function09RTYYHXKAcounterexampleW026E10
x^2 is not uniformly continuous on the real line0A8JXHYTJCcounterexampleW026A15
sin(1/x) has no limit at 00AFWQZ1PKFcounterexampleW026A15
Pairwise independent events that are not mutually independent0AR4B3K5QDcounterexampleW060A05
Z[√-5] is not a UFD0BSJYCG7NMcounterexampleW011R11
Separately continuous but not continuous function0C0KPAKK9YcounterexampleW026B05
Uncorrelated random variables that are not independent uncorrelated-does-not-imply-independent0C6J8BQXVRcounterexampleW060A05
Irreducible elements need not be prime0CHE3J7QMXcounterexampleW013A05
Absolute value is continuous but not differentiable at 00CQ9ZM383NcounterexampleW026A24
Function for which the iterated integrals differ fubini-counterexample0GJZE2E4RXcounterexampleW026B15
Curl-free field that is not conservative vortex-field0GRM3BS0DDcounterexampleW026B12
Topologist's sine curve warsaw-sine-curve0K3T4A659RcounterexampleW054G20
Non-commutativity of matrix multiplication0P506SPZP2counterexampleW015A27
Converse of Lagrange's theorem fails for the alternating group A40PDBVPP233counterexampleW020D60
A continuous nowhere differentiable function0RXK2Z808QcounterexampleW026A27
Dirichlet function0T4EJ8ZJM9counterexampleW026A15
Partial derivatives do not imply differentiability0TNZC55X3JcounterexampleW026B05
x^2 sin(1/x) is differentiable with discontinuous derivative0WDAKKN4NCcounterexampleW026A24
x^n converges pointwise but not uniformly on [0, 1]0YSRR1JTKZcounterexampleW040A30
The complex conjugate is not holomorphic z-conjugate-is-not-holomorphic0Z8ATBPM5WcounterexampleW030A99