Paper 1, Section II, E

Analysis I | Part IA, 2009

(a) What does it mean for a function f:[a,b]Rf:[a, b] \rightarrow \mathbb{R} to be Riemann integrable?

(b) Let f:[0,1]Rf:[0,1] \rightarrow \mathbb{R} be a bounded function. Suppose that for every δ>0\delta>0 there is a sequence

0a1<b1a2<b2an<bn10 \leqslant a_{1}<b_{1} \leqslant a_{2}<b_{2} \leqslant \ldots \leqslant a_{n}<b_{n} \leqslant 1

such that for each ii the function ff is Riemann integrable on the closed interval [ai,bi]\left[a_{i}, b_{i}\right], and such that i=1n(biai)1δ\sum_{i=1}^{n}\left(b_{i}-a_{i}\right) \geqslant 1-\delta. Prove that ff is Riemann integrable on [0,1][0,1].

(c) Let f:[0,1]Rf:[0,1] \rightarrow \mathbb{R} be defined as follows. We set f(x)=1f(x)=1 if xx has an infinite decimal expansion that consists of 2 s and 7 s7 \mathrm{~s} only, and otherwise we set f(x)=0f(x)=0. Prove that ff is Riemann integrable and determine 01f(x)dx\int_{0}^{1} f(x) \mathrm{d} x.

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