Paper 1, Section II, E

Let $f:[0,1] \rightarrow \mathbb{R}$ be a bounded function, and let $\mathcal{D}_{n}$ denote the dissection $0<\frac{1}{n}<\frac{2}{n}<\cdots<\frac{n-1}{n}<1$ of $[0,1]$. Prove that $f$ is Riemann integrable if and only if the difference between the upper and lower sums of $f$ with respect to the dissection $\mathcal{D}_{n}$ tends to zero as $n$ tends to infinity.

Suppose that $f$ is Riemann integrable and $g: \mathbb{R} \rightarrow \mathbb{R}$ is continuously differentiable. Prove that $g \circ f$ is Riemann integrable.

[You may use the mean value theorem provided that it is clearly stated.]

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