12.3 Continuity

Identify all discontinuities for the following functions as either a jump or a removable discontinuity.

1. ⓐ $f \left(x\right) = \frac{x^{2} - 2 x - 15}{x - 5}$
2. ⓑ $g \left(x\right) = \left\{\right. \begin{matrix}x + 1 , & x < 2 \\ - x , & x \geq 2\end{matrix}$

Identify all discontinuities for the following functions as either a jump or a removable discontinuity.

1. ⓐ $f \left(x\right) = \frac{x^{2} - 6 x}{x - 6}$
2. ⓑ $g \left(x\right) = \left\{\right. \begin{matrix}\sqrt{x} , & 0 \leq x < 4 \\ 2 x , & x \geq 4\end{matrix}$

Determine whether the function $f \left(x\right) = \left\{\right. \begin{matrix}4 x , & x \leq 3 \\ 8 + x , & x > 3\end{matrix}$ is continuous at

1. ⓐ $x = 3$
2. ⓑ $x = \frac{8}{3}$

Determine whether the function $f \left(x\right) = \left\{\right. \begin{matrix}\frac{1}{x} , & x \leq 2 \\ 9 x - 11.5 , & x > 2\end{matrix}$ is continuous at $x = 2.$

Determine whether the function $f \left(x\right) = \frac{x^{2} - 25}{x - 5}$ is continuous at $x = 5.$

Determine whether the function $f \left(x\right) = \frac{9 - x^{2}}{x^{2} - 3 x}$ is continuous at $x = 3.$ If not, state the type of discontinuity.

Determine whether the function $f$ is discontinuous for any real numbers.

Determine where the function $f \left(x\right) = \begin{cases} \frac{\pi x}{4} , x < 2 \\ \frac{\pi}{x} , \textrm{ }\textrm{ }\textrm{ }\textrm{ } 2 \leq x \leq 6 \\ 2 \pi x , x > 6 \end{cases}$ is discontinuous.

Determine whether the function below is continuous. If it is not, state the location and type of each discontinuity.

State in your own words what it means for a function $f$ to be continuous at $x = c .$

Determine why the function $f$ is discontinuous at a given point $a$ on the graph. State which condition fails.

$f \left(x\right) = \ln \left| x + 3 \right| , a = - 3$

Determine why the function $f$ is discontinuous at a given point $a$ on the graph. State which condition fails.

$f \left(x\right) = \frac{x^{2} - 16}{x + 4} , a = - 4$

Determine why the function $f$ is discontinuous at a given point $a$ on the graph. State which condition fails.

$f \left(x\right) = \begin{cases} x , x \neq 3 \\ 2 x , x = 3 \end{cases} a = 3$

Determine why the function $f$ is discontinuous at a given point $a$ on the graph. State which condition fails.

$f \left(x\right) = \left\{\right. \begin{matrix}\frac{1}{2 - x} , & x \neq 2 \\ 3 , & x = 2\end{matrix} a = 2$

Determine why the function $f$ is discontinuous at a given point $a$ on the graph. State which condition fails.

$f \left(x\right) = \left\{\right. \begin{matrix}3 + x , & x < 1 \\ x , & x = 1 \\ x^{2} , & x > 1\end{matrix} \textrm{ }\textrm{ }\textrm{ }\textrm{ } a = 1$

Determine why the function $f$ is discontinuous at a given point $a$ on the graph. State which condition fails.

$f \left(x\right) = \left\{\right. \begin{matrix}3 + 2 x , & x < 1 \\ x , & x = 1 \\ - x^{2} , & x > 1\end{matrix} \textrm{ }\textrm{ }\textrm{ }\textrm{ } a = 1$

Determine why the function $f$ is discontinuous at a given point $a$ on the graph. State which condition fails.

$f \left(x\right) = \left\{\right. \begin{matrix}\frac{x^{2} - 9}{x + 3} , & x < - 3 \\ x - 9 , & x = - 3 \\ \frac{1}{x} , & x > - 3\end{matrix} \textrm{ }\textrm{ }\textrm{ }\textrm{ } a = - 3$

Determine why the function $f$ is discontinuous at a given point $a$ on the graph. State which condition fails.

$f \left(x\right) = \frac{x^{2} - 4}{x - 2} , a = 2$

Determine why the function $f$ is discontinuous at a given point $a$ on the graph. State which condition fails.

$f \left(x\right) = \frac{x^{3} - 9 x}{x^{2} + 11 x + 24} , a = - 3$

Determine why the function $f$ is discontinuous at a given point $a$ on the graph. State which condition fails.

$f \left(x\right) = \frac{x}{\left|x\right|} , a = 0$

Determine whether or not the given function $f$ is continuous everywhere. If it is continuous everywhere it is defined, state for what range it is continuous. If it is discontinuous, state where it is discontinuous.

$f \left(x\right) = x^{3} - 2 x - 15$

Determine whether or not the given function $f$ is continuous everywhere. If it is continuous everywhere it is defined, state for what range it is continuous. If it is discontinuous, state where it is discontinuous.

$f \left(x\right) = 2 \cdot 3^{x + 4}$

Determine whether or not the given function $f$ is continuous everywhere. If it is continuous everywhere it is defined, state for what range it is continuous. If it is discontinuous, state where it is discontinuous.

$f \left(x\right) = \frac{\left|x - 2\right|}{x^{2} - 2 x}$

Determine whether or not the given function $f$ is continuous everywhere. If it is continuous everywhere it is defined, state for what range it is continuous. If it is discontinuous, state where it is discontinuous.

$f \left(x\right) = 2 x + \frac{5}{x}$

Determine whether or not the given function $f$ is continuous everywhere. If it is continuous everywhere it is defined, state for what range it is continuous. If it is discontinuous, state where it is discontinuous.

$f \left(x\right) = \ln x^{2}$

Determine whether or not the given function $f$ is continuous everywhere. If it is continuous everywhere it is defined, state for what range it is continuous. If it is discontinuous, state where it is discontinuous.

$f \left(x\right) = \sqrt{x - 4}$

Determine whether or not the given function $f$ is continuous everywhere. If it is continuous everywhere it is defined, state for what range it is continuous. If it is discontinuous, state where it is discontinuous.

$f \left(x\right) = x^{2} + \sin \left(x\right)$

$x = - 3$

$x = 4$

Evaluate $f \left(0\right) .$

What is the domain of $f \left(x\right) ?$

What condition of continuity is violated at these points?

Construct a function that passes through the origin with a constant slope of 1, with removable discontinuities at $x = - 7$ and $x = 1.$

Find the limit $\underset{x \rightarrow 1}{\lim} f \left(x\right)$ and determine if the following function is continuous at $x = 1 :$