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\centerline{MATH 133 -- Calculus with Fundamentals 1}
\centerline{Discussion Day on Continuity}
\centerline{October 2, 2015}
\vskip 10pt
\noindent
{\it Background}
\vskip 10pt
We say $f(x)$ is continuous at $x = c$ if $\lim_{x\to c} f(x) = f(c)$.  Intuitively
this means that the graph has no hole, jump, or vertical asymptote at $x = c$, although
you should be aware that this is far from a complete list of all the possible types
of \emph{discontinuities}.
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\noindent
{\it Questions}
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\begin{enumerate}
\item[(1)] Consider the function 
$$f(x) = \begin{cases}
         x^2 + 3 & {\rm if}\ x < 1\\
         -x + 10 & {\rm if}\ 1 \le x \le 2\\
         -x^2 + 6 & {\rm if}\ x > 2, x \ne 3\\
         10 & {\rm if}\ x = 3.
         \end{cases}
$$
Without using a graphing calculator, generate a plot of the graph $y = f(x)$ that
is accurate enough to let you answer the following questions.
At each of the given $x$ values, is $f$ continuous?  Why or why not?  (That is, 
if $f$ is not continuous there, what part of the definition of continuity fails?)
And if $f$ is not continuous there could you redefine the value $f(x)$ to make
a new continuous function?
\begin{enumerate}
\item[(a)]  $x = 1$
\item[(b)]  $x = 2$ 
\item[(c)]  $x = 3$.
\end{enumerate}
\item[(2)]  Which of the following quantities would you expect to be represented
by continuous functions of time and which would have one or more discontinuities?
Explain.
\begin{enumerate}
\item[(a)]  The velocity of a flying airplane.
\item[(b)]  The temperature in the middle of this room.
\item[(c)]  The balance in a bank account.
\item[(d)]  The population of Worcester.
\end{enumerate}
\item[(3)]  In 2009, the Federal income tax $T$ on on incomes $x$ up to $\$82,250$ was 
determined by the formula
$$T(x) = \begin{cases}
               0.10 x &{\rm if}\ 0 \le x \le 8350\\
               0.15x - 417.50 & {\rm if}\ 8350 \le x < 33950\\
               0.25x - 3812.50 & {\rm if}\ 33950 \le x < 82250.
\end{cases}
$$
Does $T$ have any discontinuities?  Why or why not?  Might it be advantageous in 
some situations to earn {\it less} money?
 \end{enumerate}
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