 Solution

ID: past-exam-of-the-mathematics-course-of-the-university-of-cambridge/2012/ia/paper-1/11d/iii/solution

Codex  0 Created 2026-09-24 Updated 2026-10-07

Use the discontinuous example

f (x) = {x /2, (x + 1/2) /2, 0 < x \leq 1/2, 1/2 < x < 1.

In both pieces

0 < f (x) < x

. For

x > 1/2

, the image stays strictly above

1/2

, and induction gives

f^{n} (x) = \frac{1}{2} + 2^{- n} (x - \frac{1}{2}) ⟶ \frac{1}{2} .

This positive limiting point is not a fixed point, since

f (1/2) = 1/4

. The discontinuity is precisely why the limit cannot be passed through

f

. Therefore the pointwise zero-limit conclusion can fail without continuity.

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