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Past exam of the mathematics course of the University of Cambridge / 2024 / ia / Paper 1 / 10E / c / ii / Solution Created 2026-09-24 Updated 2026-10-03

No. At

a = 0

, let

f (x) = 3 x, g (y) = y^{3} .

g

is differentiable at

f (0) = 0

, and

(g \circ f) (x) = x,

so the composite has derivative one at zero. But

\frac{f ( h ) - f ( 0 )}{h} = ∣ h ∣^{- 2/3}

is unbounded, so

f

is not differentiable. This is a nondifferentiable inner function with a differentiable nondegenerate composite.

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