Chapter 4 Functions
In this chapter we will study functions \(f: I\rightarrow {\mathbb R}\) where the domain \(I\) is a subset of \(\mathbb R\) (usually, but not always, an open interval). Recall that there are several types of intervals. Four of them are bounded:
\begin{equation*}
[a,b], (a,b],[a,b), (a,b)\text{,}
\end{equation*}
where \(a \lt b\text{,}\) and five unbounded:
\begin{equation*}
(-\infty,b], (-\infty,b), [a,\infty), (a,\infty), (-\infty,\infty)={\mathbb R}\text{.}
\end{equation*}
Four of these are open:
\begin{equation*}
(a,b), (-\infty,b), (a,\infty), (-\infty, \infty)\text{.}
\end{equation*}
Note that in Calculus and Pre-Calculus functions are often studied with domains that are not intervals. One of the simplest examples of a domain that is not an interval is the domain of \(f(x)=\frac{1}{x}\text{,}\) that is \((-\infty,0)\cup (0,\infty)\text{,}\) an interval with a single point (that is not an end point) removed. Such sets are called punctured intervals.
