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 › Continuity and Differentiation

“Continuous on a bounded interval ⇒ bounded” — false

Counterexample: f(x) = 1/x on (0,1)

Needs a compact (closed) domain.

continuity

#2 · Analysis & Linear Algebra › Continuity and Differentiation

“Uniformly continuous ⇒ Lipschitz” — false

Counterexample locked — unlock with Notes + PYQ

continuity

#3 · Analysis & Linear Algebra › Continuity and Differentiation

“Bounded and continuous on ℝ ⇒ uniformly continuous” — false

Counterexample: f(x) = sin(x²)

$x_{n} = 2 π n, y_{n} = 2 π n + π /2$ satisfy | $x_{n} - y_{n}$ | → 0 while | $f (x_{n}) - f (y_{n})$ | = 1.

continuity

#4 · Analysis & Linear Algebra › Continuity and Differentiation

“Product of uniformly continuous functions is uniformly continuous” — false

Counterexample locked — unlock with Notes + PYQ

continuity