# Section 2.6

## Continuity
:::{prf:definition}
:label: continuity
A function is continuous over an interval of its domain if its hand-drawn graph over that interval can be sketched without lifting the pencil from the paper.
:::
## Basic Graphs and Information
:::{prf:definition} Identity Function
:label: idFunction
Identity Function: $f(x)=x$.
* Domain is $\mathbb{R}$ or $(-\infty,\infty)$.
* Range is $\mathbb{R}$ or $(-\infty,\infty)$.
* Continuous everywhere or $(-\infty,\infty)$.
* Increasing everywhere or $(-\infty,\infty)$.
* ![Graph of the identity function](images/identityFunctionGraph.png)
:::
:::{prf:definition} Squaring Function
:label: sqrFunction
Squaring Function: $f(x)=x^2$.
* Domain is $\mathbb{R}$ or $(-\infty,\infty)$.
* Range is $[0,\infty)$.
* Contiuous everywhere or $(-\infty,\infty)$.
* Decreasing on $(-\infty,0)$.
* Increasing on $(0,\infty)$.
* ![Graph of the squaring function](images/sqrFunctionGraph.png)
:::

:::{plot}
images/sqrFunctionPlot.py
:::

:::{prf:definition} Cubing Function
:label: cubFunction
Cubing Function: $f(x)=x^3$
* Domain is $\mathbb{R}$ or $(-\infty,\infty)$.
* Range is $\mathbb{R}$ or $(-\infty,\infty)$.
* Continuous everywhere or $(-\infty,\infty)$.
* Increasing everywhere or $(-\infty,\infty)$.
* ![Graph of the cubing function](images/cubFunctionGraph.png)
:::
:::{prf:definition} Square Root Function
:label: sqrtFunction
Square Root Function: $f(x)=\sqrt{x}$.
* Domain is $[0,\infty)$.
* Range is $[0,\infty)$.
* Continuous on $[0,\infty)$.
* Increasing on $(0,\infty)$.
* ![Graph of the square root function](images/sqrtFunctionGraph.png)
:::
:::{prf:definition} Cube Root Function
:label: cbrtFunction
Cube Root Function: $f(x)=\sqrt[3]{x}$
* Domain is $\mathbb{R}$ or $(-\infty,\infty)$.
* Range is $\mathbb{R}$ or $(-\infty,\infty)$.
* Continuous everywhere or $(-\infty,\infty)$.
* Increasing everywhere or $(-\infty,\infty)$.
* ![Graph fo the cube root function](images/cbrtFunctionGraph.png)
:::
:::{prf:definition} Absolute Value Function
:label: absFunction
Absolute Value Function: $f(x)=|x|$.
* Domain is $\mathbb{R}$ or $(-\infty,\infty)$.
* Range is $[0,\infty)$.
* Continuous everywhere or $(-\infty,\infty)$.
* Decreasing on $(-\infty,0)$.
* Increasing on $(0,\infty)$.
* ![Graph of the the absolute value function](images/absFunctionGraph.png)
:::