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\title{MTH436 - Homework 9}
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\begin{document}
 \maketitle
 
 The solutions to problems in Section B should be submitted at the start of class on 04/24/19.
 
 
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 \section{Warm-up Questions}
 
 \begin{que}
  For the following sequences $f_n$, find $\lim_{n \to \infty} f_n$ and decide if this convergence is uniform.
  \begin{enumerate}
   \item $f \colon \R \to \R$ given by $f_n(x) = \frac{1}{n^2 + x^2}$.
   \item $f \colon [0,\infty) \to \R$ given by $f_n(x) = \frac{x^n - 1}{x^n + 1}$.
   \item $f_n \colon [-1,1] \to \R$ given by $f_n(x) = \sqrt{x^2 + \frac{1}{n^2}}$.
   \item $f_n \colon \R \to \R$ given by $f_n(x) = \frac{nx}{1+n^2x^2}$.
   \item $f_n \colon \R \to \R$ given by $f_n(x) = e^{-nx}$.
   \item $f_n \colon [0,\infty) \to \R$ given by $f_n(x) = \frac{nx}{1+nx}$.
   \end{enumerate}

 \end{que}

 \begin{que}
  Let $f_n \colon [0,1] \to \R$ given by $f_n(x) = \frac{\cos(nx)}{n}$.
  \begin{enumerate}
   \item Let $f(x) = 0$ for all $x \in [0,1]$. Prove that $f_n \unif f$ on $[0,1]$.
   \item Compute $f_n'$. Does $f_n' \to g$ for some function $g$?
  \end{enumerate}
 
 \end{que}

 
 
 \begin{que}
  Suppose $f_n \colon A \to \R$ is bounded for each $n \in \N$ (so that for each $n$ there exists $M_n \in \R$ such that $|f_n(x)| \leq M_n$ for all $x \in A$). Show that if $f_n \rightrightarrows f$ then $f$ is bounded.
 \end{que}

\begin{que}
 Let $x_1,x_2,\dotsc$ be an enumeration of the rationals in $[0,1]$. Define $f_n \colon [0,1] \to \R$ by
 \[
  f(x) = \begin{cases}
          1\qquad &\text{if $x = x_i$ for $1 \leq i \leq n$},\\
          0\qquad &\text{otherwise}.
         \end{cases}
 \]
Find $f$ such that $f_n \to f$ and discuss the integrals of $f_n$ and $f$, and points of continuity of $f_n$ and $f$.
\end{que}


 \begin{que}
   Suppose $f_n \colon A \to \R$ are continuous and $f_n \unif f$ and that $x_n \to x$ in $A$. Prove that $\lim_{n \to \infty} f_n(x_n) \to f(x)$.
  \end{que}
 
 
 
 
 
  \section{Submitted Questions}
  
  \begin{que}
   Let $f_n \colon \R \to \R$ be defined by $f_n(x) = \frac{x}{1+nx^2}$. 
   \begin{enumerate}
    \item Find $ f = \lim_{n \to \infty} f_n$.
    \item Show $f_n \rightrightarrows f$.
    \item Compute $f_n'(x)$. Is it true that $f_n' \to f'$?
   \end{enumerate}

  \end{que}

   \begin{que}
  Suppose $f_n, g_n \colon A \to \R$ such that $f_n \rightrightarrows f$ and $g_n \rightrightarrows g$.
  \begin{enumerate}
   \item Prove that $(f_n + g_n) \rightrightarrows f + g$.
   \item Prove that if $f_n$ and $g_n$ are bounded for each $n \in \N$ then $f_ng_n \rightrightarrows fg$.
   \item Give an example where $f_n \rightrightarrows f$ and $g_n \rightrightarrows g$ but $(f_ng_n)$ does not converge uniformly.
  \end{enumerate}

 \end{que}
  
  

  
  
  
  
  \section{Challenge Questions}
  
  \begin{que}
   For $x \in \R$, define the \emph{floor} of $x$ to be
   \[
    \lfloor x \rfloor = \sup \{n \in \Z \mid n \leq x \}
   \]
and the \emph{fractional part} of $x$ to be $\{x\} = x - \lfloor x \rfloor$. Consider $f \colon \R \to \R$ given by
\[
 f(x) = \sum_{n=1}^{\infty} \frac{\{nx\}}{n^2}.
\]
\begin{enumerate}
 \item Find $D$, the set of discontinuities of $f$.
 \item Show that $D$ is countable.
 \item Show that $D$ is dense in $\R$.
 \item Show that $f \in \mathcal{D}[a,b]$ for all $-\infty < a < b < \infty$.
\end{enumerate}

  \end{que}

  \begin{que}
   Define $s_d \colon [0,1) \to [0,1)$ by $s_d(x) = \{ dx \}$. For $n \in \N$, let $s_d^{\circ n} = s_d \circ s_d \circ \dotso \circ s_d$ be the $n$-fold composition of $s_d$. %Finally, for $x \in [0,1)$, define the orbit $\mathcal{O}(x)$ of $x$ to be the sequence $(f_n(x))$. Define 
   \begin{enumerate}
    \item Show that if $x \in \Q$ then $f^{\circ n}(x)$ is eventually constant.
    \item Show that if $x \in R - \Q$ then $\{ f^{\circ n}(x) \mid n \in \N \}$ is dense in $[0,1)$.
   \end{enumerate}

   
  \end{que}

  
  \end{document}
 
