Continuous Functions

Read this section for an introduction to what we mean when we say a function is continuous. Work through practice problems 1 and 2.

Definition and Meaning of Continuous

Definition of Continuity at a Point

A function $\mathbf{f}$ is continuous at $\mathbf{x}=\mathbf{a}$ if and only if $\lim\limits_{x \rightarrow a} \mathbf{f}(\mathbf{x})=\mathbf{f} \mathrm{(a)}$ .

The graph in Fig. 1 illustrates some of the different ways a function can behave at and near a point, and Table 1 contains some numerical information about the function and its behavior. the information in the table, we can conclude that $\mathrm{f}$ is continuous at 1 since $\lim\limits_{x \rightarrow 1} \mathrm{f}(\mathrm{x})=2=\mathrm{f}(1)$ .

$\mathrm{1}$	$\mathrm{f(a)}$	$\lim\limits_{x \rightarrow a} f(x)$
$\mathrm{1}$	$\mathrm{2}$	$\mathrm{2}$
$\mathrm{2}$	$\mathrm{2}$	$\mathrm{2}$
$\mathrm{3}$	$\mathrm{2}$	does not exist
$\mathrm{4}$	undefined	$\mathrm{2}$

We can also conclude from the information in the table that $\mathrm{f}$ is not continuous at 2 or 3 or 4 , because $\lim\limits_{x \rightarrow 2} \mathrm{f}(\mathrm{x}) \neq \mathrm{f}(2), \lim\limits_{x \rightarrow 3} \mathrm{f}(\mathrm{x}) \neq \mathrm{f}(3)$ , and $\lim\limits_{x \rightarrow 4} \mathrm{f}(\mathrm{x}) \neq \mathrm{f}(4)$ .