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\begin{document}


\title{On-line Math 21}


\email{david.johnson@lehigh.edu}


\urladdr{http://www.lehigh.edu/\textasciitilde{}dlj0}

\maketitle

\subsection*{Theoretical Properties Of continuous functions}

There are two basic facts about continuous functions that we should spell out
here. They are not really  astounding, but to prove them precisely is not as
trivial as it may seem.  

\begin{thm}
If \( f \) is continuous at \( b \) and \[
\lim _{x\rightarrow a}g(x)=b,\]
 then \[
\lim _{x\rightarrow a}f(g(x))=f(b)=f\left( \lim _{x\rightarrow a}g(x)\right) .\]

\end{thm}
~

\begin{thm}
If \( g \) is continuous at \( a \) and \( f \) is continuous at \( g(a) \),
then \( f\circ g(x)=f(g(x)) \) is continuous at \( a \).
\end{thm}
~

\begin{thm}
\textbf{\textup{{[}Maximum/Minimum Theorem{]}}} If \( f \) is continuous on
a closed interval \( [a,b] \), then there is a point \( x_{0}\in [a,b] \)
where \( f \) is largest (\( f(x)\leq f(x_{0}) \) for all \( x\in [a,b] \)),
and a point \( x_{1}\in [a,b] \)where \( f \) is  smallest (\( f(x)\geq f(x_{0}) \)
for all \( x\in [a,b] \)). 
\end{thm}
The hard part of this fact is that it doesn't tell you where the maximum or
minimum might be, just that they do exist.  

\begin{thm*}
\textbf{\textup{{[}Intermediate-Value Theorem{]}}} If \( f \) is continuous
on a closed interval \( [a,b] \). Let \( m \) be a number between \( f(a) \)
and \( f(b) \). Then, there is a  \( c\in [a,b] \) so that \( f(c)=m \).
 
\end{thm*}
So, for example, if the function is continuous, and at some point its value
is positive, while at another point the value is negative, then there must have
 been some point in between where the value was 0. 

\begin{thm}
If \( f(x) \) is continuous at \( x_{0} \), and \( f(x_{0})>0 \), then there
is a \( \delta >0 \) so that \( f(x)>0 \) for \( |x-x_{0}|<\delta  \). 
\end{thm}
The strength of these theorems is not really in these examples. They will be
used in the rest of this course, from time to time, as theoretical tools to
 help justify other facts. Certainly, when we get to the chapter on max/min
problems we will rely on the ``max/min'' theorem above to let us know that  a
maximum or a minimum actually does exist. 


\subsection*{Classification of discontinuities}


\subsection*{Exercises }

\begin{xca}
If \( f(x)=\cos (x) \), find the maximum and minimum of \( f \) on \( [-\pi /2,\pi ] \). 
\end{xca}

\subsubsection*{Answer}

\begin{xca}
If \( f(x)=x^{2} \), on the interval \( [-1,2) \), then does \( f \) have
a maximum? How does this relate to the theorem? 
\end{xca}

\subsubsection*{Answer}

\begin{example}
Show that \( f(x)=x^{7}+11x^{3}-15x+1 \) has at least one root in \( [0,1] \). 
\end{example}

\subsubsection*{Answer}
\end{document}
