\section{Combinations of Continuous Functions}

\begin{theorem}
	Let $A \subseteq \R$, let $f$ and $g$ be functions on $A$ to $\R$, and let $b \in \R$. Suppose that $c \in A$ and that $f$ and $g$ are continuous at $c$.
	\begin{enumerate}
		\item Then $f+g,\ f-g,\ fg$, and $bf$ are continuous at $c$.

		\item If $h:A \rightarrow \R$ is continuous at $c \in A$ and if $h(x) \neq 0$ for all $x \in A$, then the quotient $f/h$ is continuous at $c$.
	\end{enumerate}
\end{theorem}

\begin{theorem}
	Let $A \subseteq \R$, let $f$ and $g$ be continuous on $A$ to $\R$, and let $b \in \R$.
	\begin{enumerate}
		\item The functions $f+g,\ f-g,\ fg$, and $bf$ are continuous on $A$.

		\item If $h:A \rightarrow \R$ is continuous on $A$ and $h(x) \neq 0$ for $x \in A$, then the quotient $f/h$ is continuous on $A$.
	\end{enumerate}
\end{theorem}

\begin{theorem}
	Let $A \subseteq \R$, let $f:A \rightarrow \R$, and let $|f|$ be defined by $|f|(x) := |f(x)|$ for $x \in A$
	\begin{enumerate}
		\item If $f$ is continuous at at point $c \in A$, then $|f|$ is continuous at $c$.

		\item If $f$ is continuous on $A$, then $|f|$ is continuous on $A$.
	\end{enumerate}
\end{theorem}

\begin{theorem}
	Let $A \subseteq \R$, let $f:A \rightarrow \R$, and let $f(x) \geq 0$ for all $x \in A$. We let $\sqrt{f}$ be defined for $x \in A$ by $(\sqrt{f})(x) := \sqrt{f(x)}$.
	\begin{enumerate}
		\item If $f$ is continuous at at point $c \in A$, then $\sqrt{f}$ is continuous at $c$.

		\item If $f$ is continuous on $A$, then $\sqrt{f}$ is continuous on $A$.
	\end{enumerate}
\end{theorem}

\begin{theorem}
	Let $A,B \subseteq \R$ and let $f:A \rightarrow \R$ and $g:B \rightarrow \R$ be functions such that $f(A) \subseteq B$. If $f$ is continuous at a point $c \in A$ and g is continuous at $b= f(c) \in B$, then the composition $g \circ f:A \rightarrow \R$ is continuous $c$.
\end{theorem}

\begin{theorem}
	Let $A,B \subseteq \R$, let $f:A \rightarrow \R$, be continuous on $A$, and let $g:B \rightarrow \R$ be continuous on $B$. If $f(A) \subseteq B$, then the composite function $g \circ f:A \rightarrow \R$ is continuous on $A$.
\end{theorem}
