%% LyX 2.0.0 created this file.  For more info, see http://www.lyx.org/.
%% Do not edit unless you really know what you are doing.
\documentclass[english]{beamer}
\usepackage[T1]{fontenc}
\usepackage[latin9]{inputenc}
\setcounter{secnumdepth}{3}
\setcounter{tocdepth}{3}
\usepackage{amsmath}
\usepackage{amssymb}
\usepackage{graphicx}

\makeatletter
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% Textclass specific LaTeX commands.
 % this default might be overridden by plain title style
 \newcommand\makebeamertitle{\frame{\maketitle}}%
 \AtBeginDocument{
   \let\origtableofcontents=\tableofcontents
   \def\tableofcontents{\@ifnextchar[{\origtableofcontents}{\gobbletableofcontents}}
   \def\gobbletableofcontents#1{\origtableofcontents}
 }
 \long\def\lyxframe#1{\@lyxframe#1\@lyxframestop}%
 \def\@lyxframe{\@ifnextchar<{\@@lyxframe}{\@@lyxframe<*>}}%
 \def\@@lyxframe<#1>{\@ifnextchar[{\@@@lyxframe<#1>}{\@@@lyxframe<#1>[]}}
 \def\@@@lyxframe<#1>[{\@ifnextchar<{\@@@@@lyxframe<#1>[}{\@@@@lyxframe<#1>[<*>][}}
 \def\@@@@@lyxframe<#1>[#2]{\@ifnextchar[{\@@@@lyxframe<#1>[#2]}{\@@@@lyxframe<#1>[#2][]}}
 \long\def\@@@@lyxframe<#1>[#2][#3]#4\@lyxframestop#5\lyxframeend{%
   \frame<#1>[#2][#3]{\frametitle{#4}#5}}
 \def\lyxframeend{} % In case there is a superfluous frame end

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% User specified LaTeX commands.

\usepackage{appearance}

\makeatother

\usepackage{babel}
\begin{document}
%MODULE: Continuity


%USER DEFINED COMMANDS
\global\long\def\ds{{\displaystyle }}
 \global\long\def\nsp{\hspace{1em}*{9pt}}
 \global\long\def\lns{\rule{0pt}{11pt} }
 \global\long\def\xp{\mbox{\Large\sl e} }


%INPUT THE INSTRUCTIONS PAGE 


%BUILD  TITLE PAGE 



\title{Continuous functions}


\subtitle{{\hilite Continuous} functions possess a special property that
make many investigations easy.}

\makebeamertitle

\lyxframeend{}\lyxframe{{Defining the problem}}

\noindent In previous modules we have seen that
\begin{align*}
\ec\ec\lim_{x\rightarrow1}x^{2} & \ec=1^{2}=1.
\end{align*}
 Notice that you could find this answer simply by substituing $\ec x=1$
into the function $\ec f(x)=x^{2}$. Similarly, either the algebraic,
graphical, or numerical approaches will show that
\begin{align*}
\ec\lim_{x\rightarrow4}(x^{3}-4x^{2}+3) & \ec=4^{3}-4\cdot4^{2}+3=3\\
\ec\lim_{x\rightarrow0}\left|x+2\right| & \ec=\left|0+2\right|=2.
\end{align*}
Many other limits can be found simply by substituting the $x$ value.

On the other hand, many limits \emph{cannot} be found simply by substituting
the $x$ value. You have seen this with
\[
{\ec\lim_{x\rightarrow0}\sin\left(\frac{1}{x}\right)}\qquad\mathrm{or}{\ec\qquad\lim_{x\rightarrow1}\frac{x^{2}-x}{x-1}}.
\]


What we would like to know is, \emph{which} functions enjoy the property
that we can find the limit simply by susbstituting the $x$ value?

%2



\lyxframeend{}\lyxframe{{Redefining the problem}}

Let's consider the question carefully:

\noindent \begin{question}\emph{Which} functions enjoy the property
that we can find the limit simply by susbstituting the $x$ value?\end{question}


\pause{}

We can translate this into a mathematical expression:

\begin{question}Which functions $\ec f$ satisfy
\[
{\ec\lim_{x\rightarrow a}f(x)=f(a)}?
\]
\end{question}


\pause{}

We can turn this question into a definition.


\lyxframeend{}\lyxframe{{Continuity at a point}}
\begin{definition}%{}
[(Continuity at a point)] A function $\ec f$ is {\hilite continuous}
at a point $\ec x=a$ if
\[
{\ec\lim_{x\rightarrow a}f(x)=f(a)}.
\]
 That is, a function is {continuous} at $\ec x=a$ if we can find
the limit by evaluating $\ec f(a)$.
\end{definition}%{}
Geometrically, a function is continuous at $\ec x=a$ if it has no
holes, jumps, or asymptotes at $\ec x=a$. 
\begin{itemize}
\item If it has a hole at $\ec x=a$, then $\ec f(a)$ is undefined. 
\item If it has a jump at $\ec x=a$, then $\ec f(a)$ might be defined,
but the left- and right-sided limits disagree, so $\ec\lim_{x\rightarrow a}f(x)$
does not exist. 
\item If it has an asymptote at $\ec x=a$, then the limit is not finite.
(We discuss this in a subsequent module.) 
\end{itemize}
\noindent Either way, $\ec\lim_{x\rightarrow a}f(x)\neq f(a)$.


\lyxframeend{}\lyxframe{{Some examples}}

As mentioned already, $\ec f(x)=x^{2}$ is continuous at $\ec x=1$
because
\[
\ec{\lim_{x\rightarrow1}f(x)=f(1)}.
\]
 On the other hand, $\ec s(x)=\sin\left(\frac{1}{x}\right)$ is not
continuous at $\ec x=0$ because
\[
{\ec\lim_{x\rightarrow0}s(x)\neq s(0)};
\]
 in fact, $\ec s(0)$ does not even exist!

\noindent \begin{definition} If a function $\ec f$ is not continuous
at $\ec x=a$ we say that $\ec f$ is {\hilite discontinuous} at
$\ec x=a$. \end{definition}


\lyxframeend{}\lyxframe{{Which functions are continuous on their domains?}}

You should remember the following fact. \begin{theorem} The following
functions are continuous at every point in their domain: 
\begin{itemize}
\item polynomials; 
\item rational functions\\
(these have the form $\ec p(x)/q(x)$ where $\ec p,q$ are polynomials); 
\item exponential functions ($\ec a^{x}$ where $\ec a>0$ and $\ec a\neq1$); 
\item logarithmic functions ($\ec\log_{a}x$ where $\ec a>0$ and $\ec a\neq1$); 
\item trigonometric functions; 
\item sums, differences, and products of functions named above; 
\item quotients $\ec f(x)/g(x)$ of functions named above,; and 
\item composition $\ec f(g(x))$ of functions named above, \emph{provided
that the range of $\ec g$ is a subset of the domain of $\ec f$}. 
\end{itemize}
\noindent \end{theorem}


\lyxframeend{}\lyxframe{{Example}}

The following functions are continuous everywhere \emph{on their domains}. 
\begin{itemize}
\item $\ec x^{3}-4x^{2}+3$, because it is a polynomial; 
\item $\ec\left(x^{2}-x\right)/\left(x-1\right)$, because it is a rational
function; 
\item $\ec\sin(x)/(x^{2}+1)$, because it is the quotient of a trigonometric
function and a polynomial (the domain of the function is $\ec\mathbb{R}$);
and 
\item $\ec\sin(x)/(x^{2}-1)$, because it is the quotient of a trigonometric
function and a polynomial (the domain of the function is $\ec\mathbb{R}\backslash\{-1,1\}$);
and 
\item $\ec\sin\left(x^{3}-4x^{2}+3\right)$, because it is the composition
of a trigonometric function and a polynomial, \emph{and} the range
of $\ec x^{3}-4x+3$ is a subset of the domain of $\ec\sin(x)$. 
\end{itemize}

\lyxframeend{}\lyxframe{\noindent {Isn't that everything?}}

\noindent Not all functions are continuous everywhere. An important
function that you should know is the {\hilite greatest integer function},
also called the {\hilite floor function},
\[
{\ec\lfloor x\rfloor=\mathrm{the\ largest\ integer\ less\ than\ or\ equal\ to\ }x}.
\]
 For example,
\[
{\ec\lfloor3.9\rfloor=3}\qquad\mathrm{and}\qquad{\ec\lfloor-3.9\rfloor=-4}.
\]



\pause{}

\noindent  (You evaluate $\ec\lfloor a\rfloor$ in SAGE for any value
of $\ec a$ using the command \texttt{\ttcolor floor(a)}.)


\pause{}

Why is the floor function discontinuous? Estimate numerically $\ec\lim_{x\rightarrow0^{+}}\lfloor x\rfloor$
and $\ec\lim_{x\rightarrow0^{-}}\lfloor x\rfloor$. You will find
that the limits from each side are different. This means that \emph{the
limit does not exist, even though the $y$ value exists!} That makes
it impossible for $\ec\lim_{x\rightarrow0}\lfloor x\rfloor=\lfloor0\rfloor$.


\pause{}

The domain of the floor function is $\mathbb{R}$, so the floor function
is \emph{not} continuous everywhere on its domain.


\lyxframeend{}\lyxframe{{SAGE and discontinuous functions}}
\begin{overlayarea}
{\textwidth}{6in}

Discontinuous functions are a weakness for SAGE (and for every computer
algebra system). SAGE is not preprogrammed to look for discontinuities
when you try to plot its graph.


\noindent \bigskip{}

\begin{onlyenv}%{}
<1>

\noindent The following is the output of the command \texttt{\ttcolor
plot(floor(x),-3,3,thickness=2)}:


\noindent 
\begin{figure}[h!]
 

\centering{}\includegraphics[scale=0.2]{figures/floorx} 
\end{figure}


\end{onlyenv}%{}

\pause{}
\begin{onlyenv}%{}
<2>

\noindent A more accurate graph would look like this:


\begin{figure}[h!]
\centering{}\includegraphics[scale=0.2]{figures/floorx_good} 
\end{figure}



If a function is discontinuous, it is \emph{your responsibility} to
notice this, and to take the necessary precautions for obtaining a
correct graph, limit, etc.

\end{onlyenv}%{}
\end{overlayarea}

\lyxframeend{}\lyxframe{{Continuous functions}}

If a function is continuous at \emph{all} points in an interval $\ec I$,
we say that the function is {\hilite continuous on $I$}.


\pause{}

\bigskip{}
For example,
\begin{itemize}
\item polynomials are continuous on $\ec\mathbb{R}=\left(-\infty,\infty\right)$; 
\item the function $\ec1/x$ is continuous on $\ec\mathbb{R}\backslash\left\{ 0\right\} =\left(-\infty,0\right)\cup\left(0,\infty\right)$;
and 
\item the function $\ec\tan x$ is continuous on 
\[
{\ec\left\{ x\neq\frac{\pi}{2}+\pi k:\; k\in\mathbb{N}\right\} =\cdots\cup\left(-\frac{\pi}{2},\frac{\pi}{2}\right)\cup\left(\frac{\pi}{2},\frac{3\pi}{2}\right)\cup\cdots}.
\]
 
\end{itemize}
\noindent You should check that these are correct.


\lyxframeend{}\lyxframe{Application: Intermediate Value Theorem}

\noindent An important application of continuity is the {\hilite
Intermediate Value Theorem}.
\begin{theorem}%{}
[(Intermediate Value Theorem)] If
\begin{itemize}
\item the function $\ec f$ is continuous on the interval $\ec[a,b]$, and 
\item $\ec f(a)\neq f(b)$, 
\end{itemize}

then 
\begin{itemize}
\item for any $y$ value $\ec C$ between $\ec f(a)$ and $\ec f(b)$, 
\item there exists a value $\ec c$ such that 

\begin{itemize}
\item $\ec a<c<b$, and 
\item $\ec f(c)=C$. 
\end{itemize}
\end{itemize}
\end{theorem}%{}

\lyxframeend{}\lyxframe{{Example of the Intermediate Value Theorem}}

\noindent To illustrate the Intermediate Value Theorem, let $\ec f(x)=x^{2}$. 
\begin{itemize}
\item Suppose $\ec a=1$ and $\ec b=2$. 
\item Since 

\begin{itemize}
\item $\ec f$ is continuous, 
\item $\ec f(a)=1\neq4=f(b)$, 
\item and $\ec\sqrt{2}$ is between $\ec1$ and $\ec4$, 
\end{itemize}
\item we can conclude that for \emph{some} value $\ec c\in\left(1,2\right)$, 
\item $\ec f(c)=\sqrt{2}$. 
\end{itemize}
We will use this fact in another module to illustrate an application
of the Intermediate Value Theorem.


\lyxframeend{}\lyxframe{{Application: Intermediate Value Theorem}}

Effectively, the Intermediate Value Theorem says that if $\ec f$
is continuous, then a particle that travels along the curve defined
by $\ec f$ does not have to {}``jump'' to move between any two
points of the curve.

\noindent \begin{center}
\bigskip{}
\emph{This is not true for discontinuous functions!}
\par\end{center}

\noindent Using the floor function, a particle has to {}``jump''
to get from the point $\ec(-0.1,-1)$ to the point $\ec(0,0)$.

\noindent 
\begin{figure}[h!]
 

\centering{}\includegraphics[scale=0.2]{figures/floorx_good} 
\end{figure}



\lyxframeend{}\lyxframe{{Conclusion}}
\begin{itemize}
\item A function is \emph{\hilite continuous} if we can compute the two-sided
limit simply by substituting the $x$ value into the function. 
\item Not all functions are continuous everywhere on their domains. The
{\hilite greatest-integer}, or {\hilite floor} function, is
an example of this. 
\item An application of continuity is the {\hilite Intermediate Value
Theorem}. We will use this in a future module. 
\end{itemize}

\lyxframeend{}
\end{document}
