If $f$ is <continuously differentiable function>[continuously differentiable], then for every finite partition the <fundamental theorem of calculus> and the <triangle inequality> give
$$
\sum_{k=1}^n|f(t_k)-f(t_{k-1})|
\leq\sum_{k=1}^n\int_{t_{k-1}}^{t_k}|f'(s)|\,ds
\leq\int_0^1|f'(s)|\,ds<\infty.
$$
Part (b) therefore implies $\lVert f\rVert<\infty$.

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Solution

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

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ID: past-exam-of-the-mathematics-course-of-the-university-of-cambridge/2026/iii/paper-202/3/c/solution