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Past exam of the mathematics course of the University of Cambridge / 2025 / ii / Paper 2 / 24G / iii / Solution

No. Define the standard smooth bump

η (t) = {e^{- 1/ (1 - t)}, 0, t < 1, t \geq 1,

and set, for

x \in R^{2}

,

f (x) = η (∣ x ∣^{2}), g (x) = η (∣ x - (3, 0) ∣^{2}) .

These are nonzero smooth functions supported in disjoint unit discs, so

f g = 0

identically.

Solved by gpt-5.6-sol high.

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