Many statements we make about functions are only true over intervals where the function is continuous. When we say a function is continuous over an interval, we basically mean that there are no breaks in the function over that interval; that is, there are no vertical asymptotes, holes, jumps, or gaps along that interval.

##### Definition2.9.1Continuity

The function $f$ is continuous at the number $a$ if and only if $\lim\limits_{x\to a}\fe{f}{x}=\fe{f}{a}\text{.}$

There are three ways that the defining property can fail to be satisfied at a given value of $a\text{.}$ To facilitate exploration of these three manners of failure, we can separate the defining property into three sub-properties.

1. $\fe{f}{a}$ must be defined

2. $\lim\limits_{x\to a}\fe{f}{x}$ must exist

3. $\lim\limits_{x\to a}\fe{f}{x}$ must equal $\fe{f}{a}$

Please note that if either Property 1 or Property 2 fails to be satisfied at a given value of $a\text{,}$ then Property 3 also fails to be satisfied at $a\text{.}$

# Subsection2.9.1Exercises

These questions refer to the function in Figure 2.9.2.

##### 1

Complete Table 2.9.3.

##### 2

State the values of $t$ at which the function $h$ is discontinuous. For each instance of discontinuity, state (by number) all of the sub-properties in Definition 2.9.1 that fail to be satisfied.