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\begin{document}
\begin{center}
{\bf \large Math 701 Problem Set 9} \vs \\
{\bf due Friday 11/22/2013} \vs \vs
\end{center}  

\noindent
{\bf Problem 1.} Suppose $f: [a,b] \to \RR$ is increasing, that is, $x \leq y$ implies $f(x) \leq f(y)$. Show that $f \in {\mathscr R}[a,b]$. (Hint: Take a partition $P$ of $[a,b]$ into equal pieces and compute $U(f,P)-L(f,P)$.) \vs

\noindent
In the next three problems the Lebesgue-Riemann integrability criterion will be useful. \vs

\noindent
{\bf Problem 2.} Recall that the {\bit characteristic function} of a set $E \subset [a,b]$ is defined by 
$$
\chi_E(x)= \begin{cases} 1 & \ \text{if} \ x \in E \\  0 & \ \text{if} \ x \notin E. \end{cases}
$$
Under what condition on $E$ is it true that $\chi_E \in {\mathscr R}[a,b]$? \vs   
  
\noindent
{\bf Problem 3.}
\begin{enumerate}
\item[(i)]
Suppose $f \in {\mathscr R}[a,b]$ and $\phi: \RR \to \RR$ is continuous. Show that $\phi \circ f \in {\mathscr R}[a,b]$. \vs
\item[(ii)]
Show that $f \in {\mathscr R}[a,b]$ implies $f^2 \in {\mathscr R}[a,b]$. \vs 
\item[(iii)] 
Show that $f^2 \in {\mathscr R}[a,b]$ does not imply $f \in {\mathscr R}[a,b]$. \vs
\item[(iv)]
Show that $f^3 \in {\mathscr R}[a,b]$ implies $f \in {\mathscr R}[a,b]$. \vs
\end{enumerate}

\noindent
{\bf Problem 4.} Suppose $f \in {\mathscr R}[a,b]$ and $f(x)>0$ for all $x \in [a,b]$. Show that $\int_a^b f >0$. \vs

\noindent
{\bf Problem 5.} If $f: \RR \to \RR$ is $C^1$ and $E \subset \RR$ has measure zero, prove that the image $f(E)$ has measure zero. Show by an example that this is no longer true if $f$ is only assumed to be continuous. \vs 


\noindent
{\bf Problem 6.} Recall that a subset of $\RR$ is {\bit meager} if it is contained in a countable union of nowhere dense sets. Show that there is a meager set $E \subset \RR$ whose complement has measure zero. Thus, in the decomposition $\RR = E \cup E^c$, both sets are small but in different meanings. (Hint: For each $n$, cover $\QQ$ by countably many open intervals of total length $<1/n$.) 

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