Paper 1, Section II, F

Analysis I | Part IA, 2019

Let f:[0,1]Rf:[0,1] \rightarrow \mathbb{R} be a bounded function. Define the upper and lower integrals of ff. What does it mean to say that ff is Riemann integrable? If ff is Riemann integrable, what is the Riemann integral 01f(x)dx\int_{0}^{1} f(x) d x ?

Which of the following functions f:[0,1]Rf:[0,1] \rightarrow \mathbb{R} are Riemann integrable? For those that are Riemann integrable, find 01f(x)dx\int_{0}^{1} f(x) d x. Justify your answers.

(i) f(x)={1 if xQ0 if xQf(x)= \begin{cases}1 & \text { if } x \in \mathbb{Q} \\ 0 & \text { if } x \notin \mathbb{Q}\end{cases}

(ii) f(x)={1 if xA0 if xAf(x)=\left\{\begin{array}{ll}1 & \text { if } x \in A \\ 0 & \text { if } x \notin A\end{array}\right.,

where A={x[0,1]:xA=\{x \in[0,1]: x has a base-3 expansion containing a 1}\};

[Hint: You may find it helpful to note, for example, that 23A\frac{2}{3} \in A as one of the base-3 expansions of 23\frac{2}{3} is 0.1222.]\left.0.1222 \ldots .\right]

(iii) f(x)={1 if xB0 if xBf(x)=\left\{\begin{array}{ll}1 & \text { if } x \in B \\ 0 & \text { if } x \notin B\end{array}\right.,

where B={x[0,1]:xB=\{x \in[0,1]: x has a base 3-3 expansion containing infinitely many 1 s}1 \mathrm{~s}\}.

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