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\title{Homework 8 for 18.100B/C, Fall 2010\\
Due Friday 12 November}
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%\author{Richard Melrose}
%\address{Department of Mathematics, Massachusetts Institute of Technology}
%\email{rbm@math.mit.edu}
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Since Thursday November 11 is a holiday, homework is due in 2-108 by 11AM
on Friday 12 November, or by email before 5PM on the same day.

\begin{enumerate}
\item Rudin Chap 6 No 2. Suppose $f:[a,b]\longrightarrow \mathbb{R}$ is
  continuous and non-negative, $f(x)\ge0$ for all $x\in[a,b].$ Show that if
  $\int_a^bf(x)dx=0$ then $f(x)=0$ for all $x\in[a,b].$
\item Rudin Chap 6 No 4. Suppose $f:\mathbb{R}\longrightarrow \mathbb{R}$
  is defined by $f(x)=0$ when $x$ is irrational and $f(x)=1$ when $x$ is
  rational. Show that $f\notin\mathcal{R}$ (the space of Riemann integrable
  functions) on any interval $[a,b]$ where $a<b.$ 
\item Rudin Chap 6 No 5. Suppose $f:[a,b]\longrightarrow \mathbb{R}$ is
  bounded and that $f^2\in\mathcal{R}$ does it follow that
  $f\in\mathcal{R}?$ What if $f^3\in\mathcal{R}?$
\item Rudin Chap 6 No 8. Suppose that $f:[1,\infty)\longrightarrow
  \mathbb{R}$ is non-negative and monotonic decreasing. Show that
  $\lim_{b\to\infty}\int_1^bf(x)dx$ exists (and is finite) if and only if
  $\sum\limits_{n=1}^{\infty}f(n)$ exists (and is finite).
\item Suppose that $\alpha :[a,b]\longrightarrow \mathbb{R}$ is monotonic
  increasing and $f\in\mathcal{R}(\alpha )$ is real-valued and
  Riemann-Stieltjes integrable on $[a,b].$ Show that for every $\epsilon
  >0$ there is a continuous function $g:[a,b]\longrightarrow \mathbb{R}$
  such that $\int_a^b|f-g|d\alpha <\epsilon.$ For a hint, see Rudin Chap 6 No 12.
\end{enumerate}

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