Converges uniformly to 0, derivatives cos(nx) do not converge.
Counterexample bank
Part C is won by knowing which tempting claims are false. 148 counterexamples; 85 free. The rest come with the Notes pack.
AllODEPDEabeliananalyticityanovabairecalculus of variationscanonical formscauchy-riemanncompactnesscompletenesscomplexconformalconnectednesscontinuitycontour integrationconvergencedifferentiationdistributionsfieldsfixed pointgaloisgreengroupsharmonicinjectivityintegral equationsintegrationlaurentlimsuplinear algebraliouvillelpmarkovmeasuremechanicsmetric spacesmultivariablemultivariatenumericalp-groupspermutationspolynomialspower seriesprobabilityquadratic formsreal analysisregressionresiduesringssamplingseparabilityseparationsequencesseriessingularitiesstabilitystatisticssylowtestingtopologyuniform convergencewronskianzeros
#1 · Analysis & Linear Algebra › Sequences and Series of Functions
“Pointwise limit of continuous functions is continuous” — false
Counterexample locked — unlock with Notes + PYQ
uniform convergence
#2 · Analysis & Linear Algebra › Sequences and Series of Functions
“fₙ → f uniformly ⇒ fₙ′ → f′” — false
Counterexample: fₙ(x) = sin(nx)/n
uniform convergence
#3 · Analysis & Linear Algebra › Sequences and Series of Functions
“fₙ → 0 pointwise on [0,1] ⇒ ∫₀¹ fₙ → 0” — false
Counterexample locked — unlock with Notes + PYQ
uniform convergenceintegration
#4 · Analysis & Linear Algebra › Sequences and Series of Functions
“A power series converges uniformly on its open disc of convergence” — false
Counterexample locked — unlock with Notes + PYQ
power seriesuniform convergence
#5 · Analysis & Linear Algebra › Sequences and Series of Functions
“A uniformly bounded sequence in C[0,1] has a uniformly convergent subsequence” — false
Counterexample: fₙ(x) = xⁿ
Bounded by 1, but the pointwise limit is discontinuous so no subsequence converges uniformly. Equicontinuity is the missing hypothesis.
uniform convergencecompactness