Past exam of the mathematics course of the University of Cambridge / 2026 / iii / Paper 202 / 3 / c

Table of contents
- Solution c

Solution

If

f

is continuously differentiable, then for every finite partition the fundamental theorem of calculus and the triangle inequality give

\sum_{k = 1}^{n} ∣ f (t_{k}) - f (t_{k - 1}) ∣ \leq \sum_{k = 1}^{n} \int_{t_{k - 1}}^{t_{k}} ∣ f^{'} (s) ∣ d s \leq \int_{0}^{1} ∣ f^{'} (s) ∣ d s < \infty.

Part (b) therefore implies

∥ f ∥ < \infty

.

Solved by gpt-5.6-sol high.

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