 Solution

ID: past-exam-of-the-mathematics-course-of-the-university-of-cambridge/2022/ib/paper-1/12g/a/solution

Codex  0 Created 2026-09-24 Updated 2026-09-25

Ω

star-shaped with centre

z_{0}

. For small

h

with the triangle having vertices

z_{0}, w, w + h

contained in

Ω

, the vanishing triangular integral gives

g (w + h) - g (w) = \int_{[w, w + h]} f (z) d z .

Parametrizing the final segment,

\frac{g ( w + h ) - g ( w )}{h} = \int_{0}^{1} f (w + t h) d t ⟶ f (w)

by continuity. Thus

g

g^{'} (w) = f (w) .

This is the triangle-integral construction behind Morera's theorem.

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