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\begincenter{}\endcenter







\centerline {\bf TAKE HOME TEST}   
\smallskip
\centerline {\bf Math 366}
\medskip

\nin 
We always assume $(\x, d)$ and $(\y, \rho)$ are two metric spaces. 

\nin 
{\it Definition.} A mapping $f: \x\to \y$ is continuous at $x_0\in\x$ if 
for any $\varepsilon>0$ there exists a $\delta>0$ such that 
$d(x, x_0)<\delta$ implies $\rho(f(x), f(x_0))<\varepsilon$. 


\nin 
1. Suppose $f: \x\to \y$ is a mapping. Prove the following are equivalent. 

(1) $f$ is continuous at $x_0\in\x$; 

(2) for any open set $O\subset \y$ containing $f(x_0)$ the set
$f^{-1}(O)$ is open $\x$;
 
(3) for any sequence $\{x_n\}$ in $\x$, 
$x_n \to x$ implies $f(x_n)\to f(x_0)$. 

\nin
{\it Definition.} A mapping $f: \x\to \y$ is continuous in $\x$ if 
$f$ is continuous at every $x\in \x$. 

\nin
2. Suppose $f: \x\to \y$ is a mapping. Prove the following are equivalent.
 
(1) $f$ is continuous in $\x$;
 
(2) for any open set $O\subset \y$ the set
$f^{-1}(O)$ is open in $\x$;

(3) for any closed set $C\subset \y$ the set
$f^{-1}(C)$ is closed in $\x$.
 
\medskip 

\nin 
Next we discuss properties of continuous mappings on compact metric space. 

\nin  
3. Suppose $f$ is a continuous mapping of a compact metric space 
$\x$ into a metric space $\y$. Prove $f(\x)$ is compact. 
In particular, if $\y=\Bbb R^k$, then $f(\x)$ is bounded and closed. 

\nin 
4. Suppose $f$ is a continuous function in a compact metric space 
$\x$. Set 
$$M=\sup_{x\in \x}f(x), \quad m=\inf_{x\in \x}f(x). $$  
Prove there exist points $p, q\in \x$ such that 
$f(p)=M$ and $f(q)=m$. 

\smallskip \nin 
{\it Definition.}
Suppose $f$ is a mapping: $\x\to \y$.  
$f$ is uniformly continuous in $\x$ if
for any $\varepsilon>0$ there exists a $\delta>0$ such that
$$\rho(f(x_1),f(x_2))<\varepsilon\quad\text{if}\ 
d(x_1, x_2)<\delta, \ \text{and}\ x_1, x_2\in \x.$$

\nin 
5. Suppose $f$ is a continuous mapping from a compact metric space 
$\x$ into a metric space $\y$. Then  $f$
is uniformly continuous.
\medskip 

\nin
Next we generalize the notion of uniform convergence of functions. 
We always assume $(\x, \rho)$ is a metric space. 

\nin 
{\it Definition.}
Suppose $\{f_n\}$, $n=1, 2,\cdots,$ is a sequence of 
functions in $E\subset \x$.  
Then $\{f_n\}$ converges uniformly in $E\subset\x$ to a function $f$ if for
every $\varepsilon>0$ there is an integer $N$ such that
$n\ge N$ implies
$$|f_n(x)-f(x)|<\varepsilon\quad\text{for any}\ x\in E.$$

\nin 
6. Suppose that $\{f_n\}$ is a sequence of 
continuous functions in $E\subset \x$
and that  $\{f_n\}$ converges uniformly in $E$ to a function $f$. 
Prove $f$ is continuous in $E$.  

\medskip 
\nin 
Suppose $(M, \rho)$ is a compact metric space. 
$C(M)$ denotes the collection of all continuous functions: 
$M\to \Bbb R$. Define 
$$d(u, v)=\max_{x\in M}|u(x)-v(x)|\quad\hbox{for any}\ u, v\in C(M).$$ 

\nin 
7. Prove $(C(M), d)$ is a complete metric space.  
Moreover the convergence in $(C(M), d)$ is equivalent to the 
uniform convergence. 


\smallskip 
\nin 
{\it Definition.} Suppose $\{f_n\}$ is a sequence in $C(M)$. 

(a) $\{f_n\}$ is uniformly bounded if for some $M>0$ 
there holds $|f_n(x)|\le M$ for any $x\in M$ and any integer $n$; 

(b) $\{f_n\}$ is equicontinuous if for any $\varepsilon>0$ there exists a 
$\delta(\varepsilon)>0$ such that 
$|f_n(x_1)-f_n(x_2)|<\varepsilon$ for any integer $n$ 
 and $x_1, x_2\in M$ with $\rho(x_1, x_2)<\delta$. 

\nin 
8. Prove Arzela-Ascoli Theorem: Suppose $\{f_n\}$ is a sequence in $C(M)$. 
Then $\{f_n\}$ has a convergent subsequence (with respect to the metric 
$d$ defined above) if and only if $\{f_n\}$ is 
uniformly bounded and equicontinuous. 

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