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\centerline{Math 101}
\centerline{Homework 21}


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 \noindent 1)    Let $\{x_n\}$ be a sequence of limit points of $\{y_n\}$.  Suppose $x_n\rightarrow l$.  Prove that $l$ is a limit point of $\{y_n\}$.

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  \noindent 2)   Let $\{x_n\}$ be a sequence of real numbers such that $|x_n|\not\rightarrow \infty$.  Prove that $\{x_n\}$ has a limit point.  Give an example of a sequence $\{x_n\}$ such that $x_n\not\rightarrow \infty$ and $x_n\not\rightarrow -\infty$ and $\{x_n\}$ has no limit points.

  
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  \noindent 3)   Let $\{x_n\}$ be a bounded divergent sequence. Prove that $\{x_n\}$ has at least two limit points.

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 \noindent 4)  Let  $\{x_n\}$ be a bounded sequence, and let $a=\mathrm {glb}\{\mathrm{limit \medspace points \medspace of} \{x_n\}\}$, Prove that $a$ is a limit point of $\{x_n\}$.




  
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\centerline{Homework 22}

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  \noindent 1)  Let $\lbrace a_{n}\rbrace $  be 
a sequence.  Let
 
 $A=\lbrace x\in \mathbb{R} 
\vert a_{n}\geq x$ for infinitely many 
$n\in \mathbb{N}\rbrace$ and  

$B=\lbrace x\in \mathbb{ R} 
\vert a_{n}\leq x$ for infinitely many 
$n\in \mathbb{N}\rbrace$.  

Suppose that $\mathrm{lub}(A)=\mathrm{glb}(B)$.  
Prove that $\lbrace a_{n}\rbrace $ converges. 

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   \noindent 2)     Let $f:[0,\infty)\to \mathbb{R}$ be given by $f(x)=\sqrt{x}$.  Prove that $f$ is continuous.

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 \noindent 3)  Suppose that $f(x)$ is a continuous function on $\mathbb{R}$ such that $f(q)=0$ for all $q\in \mathbb{Q}$.  Prove that $f(x)=0$ for all $x\in\mathbb{R}$.  
 
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 \noindent 4) Let $f:\mathbb{R}\rightarrow \mathbb{R}$ be continuous and let $\{x_n\}$ be a sequence such that $f(x_n)\rightarrow f(l)$.  Does it follow that $x_n\rightarrow l$?  Give a proof or a counterexample.  Also, give an example of a continuous function $f:(0,1)\rightarrow \mathbb{R}$ and a Cauchy sequence $\{x_n\}$ such that $\{f(x_n)\}$ is not Cauchy.  You don't have to prove all claims about your example(s).






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\centerline{Homework 23}

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\noindent 1)  Give a proof or a counterexample to each of the following statements.  You don't have to prove all claims about your counterexample(s).


\noindent a)  Let $f:(a,b)\to \mathbb{R}$.  If $|f|$ is continuous, then $f$ is continuous.

\noindent b)  Let $f, g: (a,b)\to \mathbb{R}$.  If $f$ and $g$ are both discontinuous at a point $p\in (a,b)$, then $f+g$ is discontinuous at $p$.

\noindent c)  Let $f:[a,b]\to \mathbb{R}$ be continuous on $(a,b)$.  Then $f$ has a maximum and a minimum on $[a,b]$.

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 \noindent 2)    Let $f:\mathbb{R}\rightarrow \mathbb{R}$ be continuous at $c$, and suppose that $f(c)>0$.  Prove that there is a $\delta>0$ such that for all $x$ such that $|x-c|<\delta$, $f(x)>0$.


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 \noindent 3)  A function $f:A\to \mathbb{R}$ is {\bf increasing} if  $\forall x$, $y\in A$ with $x\leq y$, then $f(x)\leq f(y)$.  A function $f:A\to  \mathbb{R}$ is {\bf decreasing} if $\forall x$, $y\in A$ with $x\leq y$, then $f(x)\geq f(y)$.  If $f$ is either increasing or decreasing, then we say $f$ is {\bf monotonic}.
Give a proof or counterexample to each of the following statements.  You don't have to prove all claims about your example(s).


\noindent a)  Suppose that $f, g:\mathbb{R}\to \mathbb{R}$ are both continuous and monotonic, then  $f+g$ is monotonic.

\noindent b)  Let $f, g:\mathbb{R}\to \mathbb{R}$ be continuous.  If neither $f$ nor $g$ is monotonic then $f+g$ is not monotonic.

\noindent c)  Suppose that $f:\mathbb{R}\to \mathbb{R}$ is not monotonic.  Then $f$ does not have an inverse.


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\noindent 4)      Let $a\in \mathbb{R}$, and let $f(x)=
\left\{ \begin{array}{lr}
a & \mbox{if $x=0$} \\
\frac{1}{x} & \mbox{if $x\neq 0$}
\end{array}
\right.$ 

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\noindent  Prove that $f$ is not continuous at $0$ and that $f(x)$ is continuous for all $x\not = 0$.  


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