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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*{Classification of discontinuities}

Discontinuities of a function \( f \) (points \( x \) where \( f \) is not
continuous) can be separated into several distinct categories:

\textbf{Removable~discontinuities.~~} These are places \( x_{0} \) where
\( f \) just has the wrong value, and that if you change the definition of
\( f \) at \( x_{0} \), the function would be continuous.

\vspace{0.3cm}
{\par\centering \includegraphics{discontinuity-removable.ps} \par}
\vspace{0.3cm}

\textbf{Jump-discontinuities.~~} Are points \( x_{0} \) so that the left-
and right-hand limits exist, but are unequal. The value of the function could
be either one-sided limit. 

\vspace{0.3cm}
{\par\centering \includegraphics{discontinuity-jump.ps} \par}
\vspace{0.3cm}

\textbf{Poles.~~} Also called infinite discontinuities. These are places where
the limits (one side or both) are infinite. 

\vspace{0.3cm}
{\par\centering \includegraphics{discontinuity-pole.ps} \par}
\vspace{0.3cm}

\textbf{Essential~singularities.~~} These are places where not even the one-sided
limits exist, not even as infinities. Two examples would be:\[
f(x):=\left\{ \begin{array}{ll}
1, & \textrm{if}\, x\, \textrm{is}\, \textrm{rational}\\
0, & \textrm{if}\, x\, \textrm{is}\, \textrm{irrational}
\end{array},\right. \]
which is not continuous at any point, and has no one-sided limits, so has essential
singularities at all points, and\[
g(x)=\sin (1/x),\]
 which has an essential singularity at \( x=0 \).
\end{document}
