\begin{frame} \frametitle{Continuity: Composition of Functions} \begin{block}{} If $f$ and $g$ are continuous at $a$ and $c$ is a constant, then the following functions are continuous at $a$: \begin{enumerate} \pause \item $f+g$ \pause \item $f-g$ \pause \item $c\cdot f$ \pause \item $f\cdot g$ \pause \item $\frac{f}{g}$ if $g(a) \ne 0$ \end{enumerate} \end{block} \pause\bigskip All of these can be proven from the limit laws! \pause\bigskip For example, (1) can be proven as follows: \begin{talign} \lim_{x\to a}(f+g)(x) &\mpause[1]{= \lim_{x\to a}[f(x) + g(x)]} \mpause[2]{= \lim_{x\to a} f(x) + \lim_{x\to a} g(x)}\\ &\mpause[3]{= f(a) + g(a)} \mpause[4]{= (f+g)(a)} \end{talign} \pause\pause\pause\pause\pause Thus $f+g$ is continuous at $a$. \vspace{10cm} \end{frame}