%We can add comments which you don't appear in the paper like this -- using the "percentage" symbol in the beginning

\documentclass[a4paper,12pt]{article}

\usepackage[margin=1in]{geometry}  %uncomment and play with this line to change margins in your document
\usepackage{fullpage} %For standard smaller margins; You may comment this line to change margins

\usepackage{amsthm, amsmath, amssymb} %These are standard packages which you will need in almost every writing involving a lot of math.

%When theoremstyle is plain, the contents within the theorems (as defined by \newtheorem ) are italicized.
\theoremstyle{plain}
\newtheorem{theorem}{Theorem} [section]

%When theoremstyle is definition, the contents within the theorems (as defined by \newtheorem ) are NOT italicized.
\theoremstyle{definition}
\newtheorem{definition}{Definition} [section]
\newtheorem{example}{Example} [section]

%You could explore why we include "[section]" in the above definitions of "theorem", "definition" and "example". See what happens when you do not include "[section]" or when you change it to "[subsection]". If you have more questions, just google -- "latex newtheorem section" (without quotes).


\usepackage{hyperref} %For displaying url

%opening
\title{Title}
\author{Jan Doe}

\begin{document}

\maketitle
%\date{\today}
%You can write comments that don't appear in the paper.

\section{Section Title}
In this paper, we study ...

\begin{definition}
 Maybe your paper has some definitions.
\end{definition}

\subsection{Subsection Title}

\begin{theorem}
 Maybe it has a theorem.
\end{theorem}

\begin{theorem}
 Maybe it has two.
\end{theorem}


\begin{definition}
 Maybe it has some more definitions, too.
\end{definition}

\begin {example}
Here is how you give an example.
\end {example}

Maybe you need to write things in a list:

\begin{itemize}
 \item like
\item this
\item here
\end{itemize}

Or maybe you need a list with numbers.
\begin{enumerate}
 \item like
\item that
\begin{enumerate}
\item you can even
\item have lists within lists!
\end{enumerate}

\end{enumerate}

\section{Second Section}
You can do math mode in the middle of the line, like this $x + y = 5$, or you can do 
displaymath mode to get a separate line of math, like this \[x + y = 5. \]
Although it's less preferred by some, you can also write displaymath like this $$ x + y = 5. $$

\section{Third Section}
Here are some more useful mathmode symbols
\begin{itemize}
 \item $\mathbb{Z}, \; \mathbb{N}, \; \mathbb{R}$
\item $x_{n+1}, \quad x^{5+y}$
\item $\frac{4}{z-2}, \dfrac{4}{z-2}$
\item $\epsilon, \varepsilon, \delta, \Delta$
\item $\in, \; \notin$
\item $\subset, \; \subseteq, \; \subsetneq$ 
\item $\neq, \; \leq, \; \geq $
\item $\to, \; \rightarrow, \; \leftarrow$
\end{itemize}
Here are some more... but notice that sometimes things don't look as nice as you want them to... so use displaystyle in math mode:
\begin{itemize}
\item $\int_1^\infty{x^2}  \, dx \quad$  vs. $\quad \displaystyle{\int_1^\infty{x^2}  \, dx}$
\item $\sum_{i=1}^{\infty} e^{i^2} \quad$ vs. $\quad \displaystyle{\sum_{i=1}^\infty e^{i^2}}$
\item $\lim_{n\to\infty} \frac{a_n}{n} = 0 \quad$ vs. $\quad \displaystyle{\lim_{n\to\infty} \frac{a_n}{n} = 0}$

\end{itemize}

\noindent There are lots of other things you can do within Latex! So many! 
Many of them you can figure out by googling your question and the word latex. 
Try googling ``piecewise function latex" with no quotes.\\
\section{Most useful part of the whole template}
The most useful tool for help with latex is the website 
\url{http://detexify.kirelabs.org/classify.html}. You can draw symbols and it will 
tell you the tex code for them.
Another useful online reference is \url{http://en.wikibooks.org/wiki/LaTeX/}.


\section{Math 3283W Fall 2013 Writing Project Definition and Functions}

This LaTeX file contains the definitions and theorems which are part of your writing project.  You can copy and paste from this document, and use it as a starting point to experiment with LaTeX.

\noindent\hrulefill

\begin{definition}
Let $D \subseteq \mathbb{R}$.
A function $f: D \to \mathbb{R}$ is \emph{continuous at} $c \in D$ if for every sequence $x_n$ in $D$ which converges to $c$, the sequence $f \left( x_n \right) \to f (c)$. %notice \left \right) makes parentheses, brackets, etc. fit what's in between exactly, but is unnecessary for normal height things.
In symbols, assuming $x_n \in D$ for all $n$,
$$ x_n \to c \Rightarrow f \left( x_n \right) \to f (c). $$
If $f$ is continuous at every point in its domain, we simply say $f$ is \emph{continuous}.
\end{definition}

\begin{theorem}
Let $f$ and $g$ be functions for $D$ to $\mathbb{R}$ and let $c \in D$.
Suppose that $f$ and $g$ are continuous at $c$.
Then
\begin{itemize}
\item $f+g$ and $fg$ are continuous at $c$.
\item $k \cdot f$ is continuous for any real number $k$.
\item $\frac{f}{g}$ is continuou at $c$ if at $g (c) \neq 0$.
\end{itemize}
\end{theorem}

\begin{proof}
These claims follow from the Limit Laws in Theorem 4.2.1. For example, ...
\end{proof}

\begin{theorem} %LaTeX automatically numbers things for you!
Let $f$ and $g$ be continuous real-valued functions such that the range of $f$ is contained in the domain of $g$.
Then the composition $g \circ f$ is continuous.
\end{theorem}

\noindent\hrulefill \\
You can even make things go all the way to the bottom... 
\vfill


\noindent In contrast to Google Docs, LaTeX can make just about anything look beautiful and professional.
For instance, multi-line equations:
\begin{align*} %remove the * if you want each equation to be numbered for reference.
|x-y| &\le |x|+|y| \\
&< \frac{\epsilon}{2} + \frac{\epsilon}{2}\\
& = \epsilon
\end{align*}
There are no issues with with the double-struck symbol for the set of real numbers, $\mathbb{R}$.
Notice that the alignment is kept nice.

\end{document}
