Counterexample bank

Part C is won by knowing which tempting claims are false. 148 counterexamples; 85 free. The rest come with the Notes pack.

#1 · Analysis & Linear Algebra › Integration

“Riemann integrable ⇒ continuous almost everywhere fails for… (i.e. 'bounded ⇒ integrable')” — false

Counterexample locked — unlock with Notes + PYQ

integration

#2 · Analysis & Linear Algebra › Integration

“|f| Riemann integrable ⇒ f Riemann integrable” — false

Counterexample: f = 1 on ℚ ∩ [0,1], −1 elsewhere

|f| ≡ 1 is integrable; f is discontinuous everywhere.

integration

#3 · Analysis & Linear Algebra › Integration

“A composition of Riemann integrable functions is Riemann integrable” — false

Counterexample locked — unlock with Notes + PYQ

integration

#4 · 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

#5 · Analysis & Linear Algebra › Integration

“If ∫₀^∞ f converges then f(x) → 0” — false

Counterexample: f with a spike of height n and width 2/n³ at each integer n

The total area is finite but f is unbounded, so it does not tend to 0.

integration

#6 · Analysis & Linear Algebra › Lebesgue Measure and Integration

“Pointwise convergence implies convergence of the integrals” — false

Counterexample: fₙ = n·1_{(0,1/n)} on [0,1]

$f_{n} \to 0$ pointwise but $\int f_{n} = 1$ always. Domination or monotonicity is essential.

measureintegration