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\begin{frame}
\frametitle{Continuity}

\begin{block}{}
These functions are continuous at each point of their domain:
\begin{center}
polynomials \hspace{1cm} rationals \hspace{1cm} root functions\\[1ex]
(inverse) trigonometric \hspace{1cm} exponential \hspace{1cm} logarithmic
\end{center}
\end{block}
\pause\medskip

\begin{block}{}
Inverse functions of continuous functions are continuous.
\end{block}
\pause\medskip

Recall that continuity at $a$ means that
\begin{align*}
\lim_{x\to a} f(x) = f(a)
\end{align*}
and this is \emph{direct substitution}.
\pause\medskip

\begin{exampleblock}{}
Evaluate $\lim_{x\to \pi} f(x)$ where $f(x) = \frac{\sin x}{2 + \cos x}$. \pause\\[1ex]
We know that $\sin$, $\cos$ and $2$ are continuous functions.\\\pause
Then their sum and quotient are continuous on their domain.\\\pause
The domain contains $\pi$, so: $\lim_{x\to \pi} f(x) = f(\pi) = 0/(2-1) = 0$.
\end{exampleblock}
\end{frame}