====== HW 11 ======

Ross 34.2, 34.5, 34.7


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Optional: 

Rudin: Ex 15 (Hint: use 10( c ) ), 16

and an extra one: 

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 $x=0$.