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Added all Real Analysis I homeworks from Fall 2018

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\documentclass[12pt,letterpaper]{article}
\usepackage[utf8]{inputenc}
\usepackage[english]{babel}
\usepackage[normalem]{ulem}
\usepackage{cancel}
\usepackage{amsthm}
\usepackage{amsmath}
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\usepackage{array}
\usepackage[left=2cm, right=2.5cm, top=2.5cm, bottom=2.5cm]{geometry}
\usepackage{enumitem}
\newcommand{\st}{\ \text{s.t.}\ }
\newcommand{\abs}[1]{\left\lvert #1 \right\rvert}
\newcommand{\R}{\mathbb{R}}
\newcommand{\N}{\mathbb{N}}
\newcommand{\Q}{\mathbb{Q}}
\newcommand{\C}{\mathbb{C}}
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\DeclareMathOperator{\sign}{sgn}
\newtheoremstyle{case}{}{}{}{}{}{:}{ }{}
\theoremstyle{case}
\newtheorem{case}{Case}
\theoremstyle{definition}
\newtheorem{definition}{Definition}[section]
\newtheorem{theorem}{Theorem}[section]
\newtheorem*{theorem*}{Theorem}
\newtheorem{corollary}{Corollary}[section]
\newtheorem*{corollary*}{Corollary}
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\setlist[enumerate]{font=\bfseries}
\renewcommand{\qedsymbol}{$\blacksquare$}
\author{Alexander J. Tusa}
\title{Real Analysis Homework 8}
\begin{document}
\maketitle
\begin{enumerate}
\item \textbf{Section 5.1}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%% Section 5.1 Questions %%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{enumerate}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%% Question 1 - 5.1 %%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\item[1.] Prove the Sequential Criterion 5.1.3.\\
Recall the \textit{Sequential Criterion}:
\begin{theorem*}
A function $f:A \to \R$ is continuous at the point $c \in A$ if and only if for every sequence $(x_n)$ in $A$ that converges to $c$, the sequence $(f(x_n))$ converges to $f(c)$.
\end{theorem*}
Let $\varepsilon > 0$ be given. Since we're given that $f$ is continuous, we know that by the definition of continuity, there exists a $\delta >0$ such that if $x \in A$ satisfies $|x-c|<\delta$, then $|f(x)-f(c)|<\varepsilon$. Thus, for that $\delta$, since $x_n \to c,\ \exists\ n_0 \in \N \st n \geq n_0 \implies |x_n-c|<\delta$. Thus we have
\begin{align*}
n \geq n_0 &\implies |x_n - c|<\delta \\
&\implies |f(x_n)-f(c)|<\varepsilon \\
&\implies \lim\limits_{n \to \infty} f(x_n)=f(c)
\end{align*}
Now, suppose that for every sequence $(x_n)$ in $A$ that converges to $c$, $f(x_n)$ converges to $f(c)$. We will now show that the $f$ is continuous at $c$.
\begin{proof}
We want to show that $f$ is continuous at $c$.\\
By way of contradiction, suppose that $f$ is discontinuous at $c$. Then we know that $\exists\ \varepsilon >0\ \forall\ \delta > 0\ |x-c|<\delta$ and $|f(x)-f(c)| \geq \varepsilon$.\\
Let $\delta = \frac{1}{n}$. Then we have that $x_n$ is such that $|x-c|<\frac{1}{n}$ and that $|f(x_n)-f(c)| \geq \varepsilon$.\\
$\therefore$ We have a sequence $(x_n)$ such that $\lim\limits_{n \to \infty} x_n = c$ and $\lim\limits_{n \to \infty} f(x_n) \neq f(c)$. Notice however that we have a contradiction here, and thus we have that $f$ is continuous at $c$.
\end{proof}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%% Question 5 - 5.1 %%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\item[5.] Let $f$ be defined for all $x \in \R, x \neq 2$, by $f(x)=(x^2+x-6)/(x-2)$. Can $f$ be defined at $x=2$ in such a way that $f$ is continuous at this point?\\
Since we have that $f$ is not defined at $x=2$, we know that we have to factor the numerator to evaluate the limit.\\
So, for $x \neq 2$:
\[f(x)=\frac{x^2+x-6}{x-2}=\frac{\cancel{(x-2)}(x+3)}{\cancel{x-2}}=x+3\]
Thus, if we let $f(2):=5$, then $f$ is continuous:
\[\lim\limits_{x \to 2} f(x) = \lim\limits_{x \to 2} (x+3) = 5\]
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%% Question 7 - 5.1 %%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\item[7.] Let $f:\R \to \R$ be continuous at $c$ and let $f(c)>0$. Show that there exists a neighborhood $V_\delta (c)$ of $c$ such that if $x \in V_\delta (c)$, then $f(x)>0$.\\
\begin{proof}
Let $\varepsilon = \frac{f(c)}{2} > 0$. Since we are given that $f$ is continuous at $c$, we know that there exists $\delta > 0$ such that
\[|x-c|<\delta \implies |f(x)-f(c)|<\varepsilon = \frac{f(c)}{2}\]
Thus, if we define $V_\delta (c) := (c-\delta, c+\delta)$ and if we let $x \in V_\delta (c)$, then we have
\begin{align*}
x \in V_\delta (c) &\implies |x-c|<\delta \\
&\implies |f(x)-f(c)|<\varepsilon \\
&\implies -\varepsilon < f(x)-f(c)<\varepsilon \\
&\implies f(c)-\frac{f(c)}{2} < f(x) &(\varepsilon = \frac{f(c)}{2}) \\
&\implies \frac{1}{2}f(c) < f(x) \\
&\implies f(x)>0 &(f(c)>0)
\end{align*}
Thus we have that there exists a neighborhood $V_\delta(c)$ such that if $x \in V_\delta (c)$, then $f(x)>0$.
\end{proof}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%% Question 11 - 5.1 %%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\item[11.] Let $K>0$ and let $f:\R \to \R$ satisfy the condition $|f(x)-f(y)|\leq K|x-y|$ for all $x,y \in \R$. Show that $f$ is continuous at every point $c \in \R$.\\
\begin{proof}
We want to show that $f$ is continuous at every point $c \in \R$.\\
Let $\varepsilon >0$ be given, and let $\delta = \frac{\varepsilon}{k}$. Then, for any $c \in \R$,
\begin{align*}
|x-c|<\delta &\implies |x-c|<\frac{\varepsilon}{k} \\
&\implies k|x-c| < \varepsilon \\
&\implies |f(x)-f(c)| \leq k|x-c|<\varepsilon \\
&\implies |f(x)-f(c)|<\varepsilon
\end{align*}
Thus we have that $f$ is continuous for $c \in \R$. Since $c$ is arbitrary, this also shows that $f$ is continuous for all $c \in \R$.
\end{proof}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%% Question 12 - 5.1 %%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\item[12.] Suppose that $f:\R \to \R$ is continuous on $\R$ and that $f(r)=0$ for every rational number $r$. Prove that $f(x)=0$ for all $x \in \R$.\\
\begin{proof}
We want to show that $f(x)=0,\ \forall\ x \in \R$.\\
Let $x \in \R$. Since we know that $\Q$ is dense in $\R$, we know that we can find a sequence $(x_n)$ of rational numbers such that $(x_n)$ converges to $x$. Thus, by continuity of $f$ at $x$, $(f(x_n))$ converges to $f(x)$. Since $x_n$ is rational, we have that $f(x_n)=0\ \forall\ n \in \N$. Thus we have that $f(x) = \lim f(x_n) = \lim 0 = 0$.\\
$\therefore$ We have that $f(x)=0$ for all $x \in \R$.
\end{proof}
\end{enumerate}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%% Question 2 %%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\item Use the definition of continuity to show that the given $f$ is continuous.
\begin{enumerate}
\item Let $f:\R \to \R$ be given by $f(x)=x^2$.\\
Recall the definition of continuity:
\theoremstyle{definition}
\begin{definition}
Let $A \subseteq \R$, let $f:A \to \R$, and let $c \in A$. We say that $f$ is \textbf{continuous at} $c$ if, given any number $\varepsilon > 0$, there exists $\delta >0$ such that if $x$ is any point of $A$ satisfying $|x-c|<\delta$, then $|f(x)-f(c)|<\varepsilon$. If $f$ fails to be continuous at $c$, then we say that $f$ is \textbf{discontinuous at} $c$.
\end{definition}
\begin{proof}
In order to show that $f$ is continuous for all $c \in \R$, we must consider two cases: $c\neq 0$, and $c=0$.\\
Suppose $c \neq 0$. Then we have the following:
\begin{align*}
|f(x)-f(c)| &= |x^2 - c^2| \\
&=|(x-c)(x+c)| \\
&=|x-c||x+c| \\
&< |x-c|(|x|+|c|) \\
&<\varepsilon
\end{align*}
Which yields that
\[|x-c|<\frac{\varepsilon}{|x|+|c|}< \frac{\varepsilon}{|x|}\]
So, for the case when $c \neq 0$, let $\delta=\frac{\varepsilon}{|x|}$.\\
Now, suppose that $c=0$. Then we have the following:
\begin{align*}
|f(x)-f(c)| &= |x^2-0^2| \\
&= |x^2| \\
&=(|x|)^2 \\
&< \varepsilon
\end{align*}
So, we have now that
\[(|x|)^2 < \varepsilon \implies |x|<\sqrt{\varepsilon}\]
Thus, let $\delta=\sqrt{\varepsilon}$.\\
$\therefore$ Since we have found definitions of $\delta$ that satisfy the definition of continuity, we have that $f$ is continuous $\forall c \in \R$.
\end{proof}
\item Let $f:(0,\infty) \to \R$ be given by $f(x)=1/x$. \\
\begin{proof}
Suppose $x>\frac{c}{2}$, since $c>0$. Then we have that
\[\abs{f(x)-f(c)}=\abs{\frac{1}{x}-\frac{1}{c}}=\frac{|x-c|}{xc}<\frac{2}{c^2}|x-c|<\varepsilon \implies |x-c|<\frac{c^2\varepsilon}{2}\]
So, if we let $\delta < \min \{\frac{c^2\varepsilon}{2}, \frac{c}{2}\}$, then we have that if $|x-c|<\delta,\ x>\frac{c}{2}$, and $\abs{\frac{1}{x}-\frac{1}{c}}<\frac{2}{c^2}\delta \leq \varepsilon$.\\
$\therefore$ We have that $f$ is continuous $\forall\ c \in (0,\infty)$.
\end{proof}
\item Let $f:\R \to \R$ be given by $f(x)=|x|$ (This is Problem 10 in Section 5.1)\\
\begin{proof}
We must first note the following:
\begin{align*}
|a| &= |a-b+b| \\
&\leq |a-b|+|b| \\
&\Downarrow \\
|a|-|b| &\leq |a-b|
\end{align*}
$\implies \pm(|a|-|b|) \leq |a-b|$
We also note:
\begin{align*}
|b| &= |b-a+a| \\
&\leq |b-a| + |a| \\
&=|a-b|+|a| \\
&\Downarrow \\
|b|-|a| &\leq |a-b|
\end{align*}
$\implies ||a|-|b|| \leq |a-b|$
\begin{align*}
|f(x)-f(c)| &= ||x|-|c|| \\
&\leq |x-c| \\
&<\varepsilon
\end{align*}
Thus if we let $\varepsilon = \delta$, then we have that $|x-c|<\delta$.\\
$\therefore$ We have that $f(x)$ is continuous $\forall\ c \in \R$.
\end{proof}
\end{enumerate}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%% Question 3 %%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\item Let $f(x)=\begin{cases}
3x+2 & \text{if } x \text{ is rational} \\
6-x & \text{if } x \text{ is irrational}
\end{cases}$
\begin{enumerate}
\item Determine whether or not $f$ is continuous at $x=1$.\\
$f$ is continuous at $x=1$ since we proved in $HW 7$ that $\lim\limits_{x \to 1} f(x)$ exists, $\lim\limits_{x \to 1} f(x)=f(1)=5$.
\item Determine whether or not $f$ is continuous at $x=0$.\\
$f$ is not continuous for $x \neq 1$ (and not $x = 0$) since from Homework 7 again, $\lim\limits_{x \to c} f(x)$ does not exists for $x \neq 1$.
\end{enumerate}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%% Question 4 %%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\item Let $g(x)=\begin{cases}
2x & \text{if } x \text{ is rational} \\
x+3 & \text{if } x \text{ is irrational}
\end{cases}$ \\
Find all points where $g$ is continuous (This is Problem 13 of Section 5.1)\\
First, let $c$ be some point of continuity of $g$. Then since both $\Q$ and $\R \setminus \Q$ are dense in $\R$, we can find sequences $(x_n) \subset \Q$ and $(y_n) \subset \R \setminus \Q$ such that $(x_n) \to c$ and $(y_n) \to c$. Then we have the following:
\begin{align*}
g(c) &= \lim\limits_{n \to \infty} g(x_n) \\
&= \lim\limits_{x \to \infty} 2x_n \\
&= 2c &(x_n \to c) \\
\\
g(c) &= \lim\limits_{n \to \infty} g(y_n) \\
&= \lim\limits_{n \to \infty} y_n + 3 \\
&= c+3 &(y_n \to c) \\
\\
\implies 2c &= c+3 \\
&\Updownarrow \\
c &= 3
\end{align*}
Thus, we have that if $g$ is continuous at a point $c$, then it must be $c=3$. Thus we conjecture that $g$ is continuous only at $3$.\\
\begin{proof}
\begin{align*}
|g(x)-g(3)| &= |g(x)-6| &(3 \in \Q) \\
&\leq \sup \{|2x-6|,|(x+3)-6|\} &(x \text{ is either rational or irrational}) \\
&= \sup \{2|x-3|,|x-3|\} \\
&=2|x-3|
\end{align*}
Thus $\forall\ \varepsilon >0$ let $\delta=\frac{\varepsilon}{2}$. Then we have that
\[|x-3| < \delta \implies |g(x)-g(3)|<\varepsilon\]
$\therefore$ We have that $g$ is only continuous at $c=3$.
\end{proof}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%% Question 5 %%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\item Give an example of the following, if possible.
\begin{enumerate}
\item A function $f$ defined on $\R$ such that it is not continuous at any point in $\R$.\\
Consider the Dirichlet function. This function is not continuous at any points.
\item A function $f$ defined on $\R$ such that it is continuous at exactly one point in $\R$.\\
Consider the function $f(x):= \begin{cases}
x, & \text{if } x \in \Q \\
0, & \text{if } x \in \R \setminus \Q
\end{cases}$ \\
Notice that this function is continuous only at $x=0$.
\item A function $f$ defined on $\R$ such that it is continuous at exactly two points in $\R$.\\
Consider the function $f(x):=\begin{cases}
x^2, & \text{if } x \in \Q \\
1, &\text{if } x \notin \Q
\end{cases}$ \\
We notice that this function is continuous only at the points $-1$ and $1$.
\end{enumerate}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%% Section 5.2 %%%%%%%%
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\item \textbf{Section 5.2}
\begin{enumerate}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%% Question 2 - 5.2 %%%%%%%%
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\item[2.] Show that if $f:A \to \R$ is continuous on $A \subseteq \R$ and if $n \in \N$, then the function $f^n$ defined by $f^n(x)=(f(x))^n$, for $x \in A$, is continuous on $A$.\\
\begin{proof}
We want to show that $f^n$ is continuous. We prove it by method of mathematical induction.\\
\textbf{Basis Step:} Let $n=1$. Then we have that $f^1(x)=(f(x))^1=f(x) \implies f^1=f$. Since we are given that $f$ is continuous, we know that $f^1$ is continuous as well.\\
\textbf{Inductive Step:} Assume that $f^n(x)=(f(x))^n$ is continuous $\forall\ n \in \N$.\\
\textbf{Show:} We want to show that $f^{n+1}(x)=(f(x))^{n+1}$ is continuous $\forall\ n \in \N$. Then we have
\[f^{n+1}=f^n \cdot f\]
Recall \textit{Theorem 5.2.1}:
\begin{theorem*}
Let $A \subseteq \R$, let $f$ and $g$ be functions on $A$ to $\R$, and let $b \in \R$. Suppose that $c \in A$ and that $f$ and $g$ are continuous at $c$.
\begin{enumerate}
\item Then $f+g,\ f-g,\ fg$, and $bf$ are continuous at $c$.
\item If $h:A \rightarrow \R$ is continuous at $c \in A$ and if $h(x) \neq 0$ for all $x \in A$, then the quotient $f/h$ is continuous at $c$.
\end{enumerate}
\end{theorem*}
Since both $f^n$ and $f$ are continuous functions, by \textit{Theorem 5.2.1}, we have that $f^n\cdot f=f^{n+1}$ is a continuous function on $A$ since it is the product of continuous functions on $A$.\\
$\therefore$ By the Principle of Mathematical Induction, we have that $f^n$ is continuous $\forall\ n \in \N$ on $A$.
\end{proof}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%% Question 3 - 5.2 %%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\item[3.] Give an example of functions $f$ and $g$ that are both discontinuous at a point $c$ in $\R$ such that $(a)$ the sum $f+g$ is continuous at $c$ and $(b)$ the product $fg$ is continuous at $c$.\\
Consider the functions $f(x):=\begin{cases}
1, & x=0 \\
0, & x \neq 0
\end{cases}$, and $g(x):=\begin{cases}
0, & x=0 \\
1, & x \neq 0
\end{cases}$ \\
We first note that both $f(x)$ and $g(x)$ are discontinuous at $x=0$.
\begin{enumerate}
\item[(a)] Note that the sum
\[f(x)+g(x):=\begin{cases}
1+0, & x=0 \\
0+1, & x \neq 0
\end{cases} = \begin{cases}
1, & x=0 \\
1, & x \neq 0
\end{cases}=1\].
Thus, we have that the sum $f(x)+g(x)$ is a constant function, and thus is defined $\forall\ x \in \R$, and thus $f(x)+g(x)$ is continuous at $c$, where $c=0$.
\item[(b)] Now, the product $f(x)\cdot g(x)$ is
\[f(x) \cdot g(x):= \begin{cases}
1 \cdot 0, & x=0 \\
0 \cdot 1, & x \neq 0
\end{cases} = \begin{cases}
0, & x=0 \\
0, & x \neq 0
\end{cases}=0\]
Thus we have that the product of $f(x) \cdot g(x)$ is also a continuous function since $f(x) \cdot g(x)=0$. Thus, we have that $f(x) \cdot g(x)$ is continuous $\forall\ x \in \R$. Hence $f(x) \cdot g(x)$ is continuous at $c$, where $c=0$.
\end{enumerate}
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%%%%%%%% Question 5 - 5.2 %%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\item[5.] Let $g$ be defined on $\R$ by $g(1):=0$, and $g(x):=2$ if $x \neq 1$, and let $f(x):=x+1$ for all $x \in \R$. Show that $\lim\limits_{x \to 0} g \circ f \neq (g \circ f)(0)$. Why doesn't this contradict Theorem 5.2.6?\\
Recall \textit{Theorem 5.2.6}:
\begin{theorem*}
Let $A,B \subseteq \R$ and let $f:A \rightarrow \R$ and $g:B \rightarrow \R$ be functions such that $f(A) \subseteq B$. If $f$ is continuous at a point $c \in A$ and g is continuous at $b= f(c) \in B$, then the composition $g \circ f:A \rightarrow \R$ is continuous $c$.
\end{theorem*}
Note that we have $g(x):=\begin{cases}
0, & x=1 \\
2, & x \neq 1
\end{cases}$, and $f(x):=x+1$.\\
Thus $(g \circ f)(x)=g(f(x))=g(x+1)=\begin{cases}
0, & x+1=1 \\
2, & x+1 \neq 1
\end{cases} = \begin{cases}
0, & x=0 \\
2, & x \neq 0
\end{cases}$\\
This gives us that $\lim\limits_{x \to 0} (g \circ f)(x) = \lim\limits_{x \to 0} 2 = 2$, since $x \to 0 \implies x \neq 0$\\
Thus we have that $\lim\limits_{x \to 0} (g \circ f)(x) \neq (g \circ f)(0)$ since $2 \neq 0$.\\
The reason that this does not violate \textit{Theorem 5.2.6} is because $g$ is discontinuous at $f(0)$, since $f(0)=1$. \\
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%% Question 7 - 5.2 %%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\item[7.] Give an example of a function $f:[0,1] \to \R$ that is discontinuous at every point of $[0,1]$ but such that $|f|$ is continuous on $[0,1]$.\\
Let $f(x):=\begin{cases}
-1, & x \in \Q \\
1, & x \in \R \setminus \Q
\end{cases}$\\
Recall \textit{Theorem 2.4.8 - The Density Theorem}:
\begin{theorem*}[\textbf{The Density Theorem}]
If $x$ and $y$ are any real numbers with $x<y$, then there exists a rational number $r \in \Q$ such that $x < r < y$.
\end{theorem*}
And also recall \textit{Corollary 2.4.9}:
\begin{corollary*}
If $x$ and $y$ are real numbers with $x < y$, then there exists an irrational number $z$ such that $x < z < y$.
\end{corollary*}
Recall the \textit{Discontinuity Criterion}:
\begin{theorem*}[\textbf{Discontinuity Criterion}]
Let $A \subseteq \R$, let $F:A \rightarrow \R$, and let $c \in A$. Then $f$ is discontinuous at $c$ if and only if there exists a sequence $(x_n)$ in $A$ such that $(x_n)$ converges to $c$, but the sequence $(f(x_n))$ does not converge to $f(c)$.
\end{theorem*}
Consider the sequence $x_n:=(\frac{1}{3}, \frac{1}{3}, \dots, \frac{1}{3})$. Clearly $x_n \in [0,1]\ \forall\ n \in \N$. However, note that $\lim\limits_{x \to \frac{1}{3}} (x_n) = \frac{1}{3}$, but $\lim\limits_{x \to \frac{1}{3}} (f(x_n)) = 1$ but $f(\frac{1}{3})=-1$. Thus we have that $\lim\limits_{x \to \frac{1}{3}} (f(x_n)) \neq f(c)$, where $c=\frac{1}{3}$. Thus by the \textit{Discontinuity Criterion}, since there exists a sequence $(x_n) \in [0,1]$ such that $(x_n) \to c \in [0,1]$ but $(f(x_n)) \nrightarrow f(c)$. Thus we have that $f$ is discontinuous at $c=\frac{1}{3}$. However, note that by the \textit{Density Theorem} and by \textit{Corollary 2.4.9}, we have that $f$ is discontinuous at $c\ \forall c \in [0,1]$.\\
However, note that $|f(x)|=1\ \forall\ x \in [0,1]$. Thus we have that $|f|$ is continuous $\forall\ c \in [0,1]$.\\
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%% Question 8 - 5.2 %%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\item[8.] Let $f,g$ be continuous from $\R$ to $\R$ and suppose that $f(r)=g(r)$ for all rational numbers $r$. Is it true that $f(x)=g(x)$ for all $x \in \R$?\\
This is actually true.
\begin{proof}
Let $x \in \R$ be arbitrary. By the \textit{Density Theorem}, we know that we can find a sequence $(x_n) \subset \Q \st (x_n) \to x$.\\
Then we have:
\begin{align*}
f(x) &= \lim\limits_{n \to \infty} f(x_n) &(f\text{ is continuous}) \\
&= \lim\limits_{n \to \infty} g(x_n) &(x_n \in \Q) \\
&= g(x) &(g\text{ is continuous}) \\
\implies f(x) &= g(x)\ \forall\ x \in \R
\end{align*}
\end{proof}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%% Question 14 - 5.2 %%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\item[14.] Let $g:\R \to \R$ satisfy the relation $g(x+y)=g(x)g(y)$ for all $x,y \in \R$. Show that if $g$ is continuous at $x=0$, then $g$ is continuous at every point of $\R$. Also if we have $g(a)=0$ for some $a \in \R$, then $g(x)=0$ for all $x \in \R$.\\
\begin{proof}
We first note that since we're given that $g$ is continuous at $x=0$, we have
\begin{align*}
g(0) &= g(0+0) \\
&=g(0) \cdot g(0) &(g(x+y)=g(x)g(y),\ \forall x,y \in \R) \\
&=g(0)^2
\end{align*}
This tells us that we have two cases to consider for $g(0)$: $g(0)=0$ or $g(0)=1$.\\
\begin{case}{$g(0)=0$}
Let $g(a)=0$ for some $a \in \R$. Then we have that $\forall\ x \in \R$,
\[g(x) = g((x-a)+a)=g(x-a) \cdot g(a)=0\]
Thus, if $g(a)=0\ \forall\ a \in \R$, then $g \equiv 0$ and thus $g$ is constant. Thus, $g$ is continuous $\forall\ x \in \R$.
\end{case}
\begin{case}{$g(0)=1$}
Let $g(c) \neq 0\ \forall\ c \in \R$. Otherwise we have that $g$ is a zero function, which was shown to be continuous as the result of \textit{Case 1}, and this contradicts the fact that we let $g(0)=1 \neq 0$.\\
Let $\varepsilon > 0$ be given, and let $c \in \R$ be arbitrary such that $c \neq 0$. Since $g$ is continuous at 0 and $g(c) \neq 0$, we know that $\exists\ \delta >0 \st \forall\ x \in \R$, we have:
\[|x-0|<\delta \implies |g(x)-g(0)|<\frac{\varepsilon}{|g(c)|}\]
that is, since $g(0)=1$,
\[|x| < \delta \implies |g(x)-1|<\frac{\varepsilon}{|g(c)|}\]
Thus, for $x \in \R \st |x-c|<\delta$, we have
\begin{align*}
|g(x)-g(c)| &= |g((x-c)+c)-g(c)| \\
&=|g(x-c)g(c)-g(c)| &(g(a+b)=g(a)g(b)) \\
&= |g(c)(g(x-c)-1)| \\
&=|g(c)||g(x-c)-1| \\
&= |g(c)||g(h)-1| &(\text{Substitue } h:=x-c) \\
&<|g(c)|\frac{\varepsilon}{|g(c)|} &(|h|=|x-c|<\delta) \\
&= \varepsilon
\end{align*}
\end{case}
$\therefore$ We have that $g$ is continuous at $c$. Since $c \neq 0$ was arbitrary and $g$ is continuous at 0, we have that $g$ is continuous on $\R$.
\end{proof}
\end{enumerate}
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%%%%%%%% Question 7 %%%%%%%%
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\item Prove or justify if true. Provide a counterexample if false.
\begin{enumerate}
\item Let $f$ be defined on $[a,b]$. Let $x_n$ be any sequence in $[a,b]$. The sequence $\{f(x_n)\}$ converges to $f(c)$ for $x_n$ converging to $c \in [a,b]$ implies that $f$ is continuous at $x=c$. \\
This is true. Recall the \textit{Sequential Criterion for Continuity}:
\begin{theorem*}[\textbf{Sequential Criterion for Continuity}]
A function $f:A \rightarrow \R$ is continuous at the point $c \in A$ if and only if for every sequence $(x_n)$ in $A$ that converges to $c$, the sequence $(f(x_n))$ converges to $f(c)$.
\end{theorem*}
This is true by the \textit{Sequential Criterion for Continuity}.\\
\item If $f$ is continuous on $D$ and the sequence $x_n$ in $D$ is a converging sequence, then the sequence $\{f(x_n)\}$ converges.\\
This statement is false. Consider the function $f:(0,1) \to \Q$ given by $f(x)=\frac{1}{x}$, and consider the sequence $x_n:= (0.1, 0.01,0.001,0.0001, \dots),\ \forall\ n \in \N$. Then we have that $x_n \to 0$, but since $0 \notin (0,1)$ and since $(0,1)$ is not a compact set, this violates the \textit{Sequential Criterion for Continuity} and thus this statement is false.
\end{enumerate}
\end{enumerate}
\end{document}