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Counterexample bank

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

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#1 · Complex Analysis, Algebra & Topology › Zeros and Mappings

“f′(z) ≠ 0 everywhere implies f is injective” — false

Counterexample: f(z) = eᶻ on ℂ

f′ = eᶻ never vanishes, yet $f (z) = f (z + 2 π i)$ . Non-vanishing derivative gives only local injectivity.

complexinjectivity