continuity-rules.tex

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\begin{document}
\large




\noindent we have 
\begin{description}

\item[Rule \#0 (polynomial)] Every polynomial is continuous everywhere.

\item[Rule \#1 (rational)] Every rational function is continuous everywhere it's defined.

\item[Rule \#2]  Each of the following functions are continuous everywhere they
are defined: power (both integer and noninteger powers), trigonometric, inverse trigonometric, exponential, and logarithmic.

\item[Rule \#3] Let $F$ and $G$ be functions that are continuous at $c$ and let
$a,b$ be numbers and let $n$ be a positive integer; each of the following are 
continuous at $c$:
\begin{align*}
    &a F + b G \\
    &F G & \\
    &F/G & (\mbox{ provided  $G(c) \neq 0$) \\
    &F^n & \\
    &F^{1/n} & (\mbox{ provided $F{1/n}$ is defined on a neighborhood of $c$})
\end{align*}

\item[Rule \#4] Let $G$ be continuous at $c$ and let $F$ be continuous at $F(c)$.
Then $F \circ G$ is continuous at $c$.

\end{description}
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