Ross 34.2, 34.5, 34.7

Extra: Let $f:[0,1] \to \R$ be given by $$ f(x) = \begin{cases} 0 &\text{if } x = 0 \cr \sin(1/x) &\text{if } x \in (0,1] \end{cases}. $$ And let $\alpha: [0, 1] \to \R$ be given by $$ \alpha(x) = \begin{cases} 0 &\text{if } x = 0 \cr \sum_{n \in \N, 1/n<x} 2^{-n} &\text{if } x \in (0,1] \end{cases}. $$ Prove that $f$ is integrable with respect to $\alpha$ on $[0,1]$. Hint: prove that $\alpha(x)$ is continuous at $x=0$.