Paper 1, Section II, E

Analysis I | Part IA, 2015

Let f:[0,1]Rf:[0,1] \rightarrow \mathbb{R} be a bounded function, and let Dn\mathcal{D}_{n} denote the dissection 0<1n<2n<<n1n<10<\frac{1}{n}<\frac{2}{n}<\cdots<\frac{n-1}{n}<1 of [0,1][0,1]. Prove that ff is Riemann integrable if and only if the difference between the upper and lower sums of ff with respect to the dissection Dn\mathcal{D}_{n} tends to zero as nn tends to infinity.

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

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

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