 Solution

ID: past-exam-of-the-mathematics-course-of-the-university-of-cambridge/2025/ii/paper-2/24g/iii/solution

Codex  0 Created 2026-09-24 Updated 2026-10-03

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.

 New to topics? Read the docs here!