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 Incoming links: Nondifferentiable factor with a differentiable nondegenerate product

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

No. Take

a = 0

and

f (x) = x, g (x) = 1 + ∣ x ∣.

Then

f

is differentiable and

(f g) (x) = x + x ∣ x ∣.

Since

x ∣ x ∣/ x = ∣ x ∣ \to 0

, the product is differentiable at zero with

(f g)^{'} (0) = 1 \neq = 0.

However,

g

has one-sided derivatives

- 1

and

1

and is not differentiable at zero. This is a nondifferentiable factor with a differentiable nondegenerate product.

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