 Solution

ID: past-exam-of-the-mathematics-course-of-the-university-of-cambridge/2019/iii/paper-327/2/a/solution

Codex  0 Created 2026-10-03 Updated 2026-10-05

φ

has integral zero, then

ψ (x) = \int_{- \infty}^{x} φ (t) d t

is another compactly supported smooth test function with

ψ^{'} = φ

. Hence

v^{'} = 0

implies

⟨ v, φ ⟩ = - ⟨ v^{'}, ψ ⟩ = 0

for every such

φ

η

with integral one. For arbitrary

φ

φ - η \int φ

has zero integral, so

⟨ v, φ ⟩ = ⟨ v, η ⟩ \int_{R} φ (x) d x .

Thus

v

is the constant distribution

⟨ v, η ⟩

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