Chapter 14, Problem 12CT

(a)

To determine

Whether $f$ is continuous at $x=-2$ by using the graph of $y=f\left(x\right)$. If it is not continuous then explain why not.

The graph of $y=f\left(x\right)$ is,

Answer to Problem 12CT

Solution:

$f$ is continuous at $x=-2$.

Explanation of Solution

Given information:

The graph of $y=f\left(x\right)$:

Explanation:

By observing the graph as $x$ approaches to $-2$ from the left hand side the value of function approaches to $2$ and as $x$ approaches to $-2$ from the right hand side the value of function approaches to $2$ and at $x=-2$ the value of function is $2$.

A function is continuous at $x=c$ when $\underset{x\to c^{-}}{\lim }f\left(x\right)=\underset{x\to c^{+}}{\lim }f\left(x\right)=f\left(c\right)$.

Here, $\underset{x\to -2^{-}}{\lim }f\left(x\right)=\underset{x\to -2^{+}}{\lim }f\left(x\right)=f\left(2\right)=2$.

So, by using the definition of continuity, $f$ is continuous.

Therefore, $f$ is continuous at $x=-2$.

(b)

To determine

Whether $f$ is continuous at $x=1$ by using the graph of $y=f\left(x\right)$. If it is not continuous then explain why not.

The graph of $y=f\left(x\right)$ is,

Answer to Problem 12CT

Solution:

$f$ is not continuous at $x=1$.

Explanation of Solution

Given information:

The graph of $y=f\left(x\right)$:

Explanation:

By observing the graph as $x$ approaches to $1$ from the left hand side the value of function approaches to $2$ and as $x$ approaches to $1$ from the right hand side the value of function approaches to $2$ and at $x=1$ the value of function is $1$.

A function is continuous at $x=c$ when $\underset{x\to c^{-}}{\lim }f\left(x\right)=\underset{x\to c^{+}}{\lim }f\left(x\right)=f\left(c\right)$.

Here, $\underset{x\to 1^{-}}{\lim }f\left(x\right)=\underset{x\to 1^{+}}{\lim }f\left(x\right)=2\ne f\left(1\right)$.

So, by using the definition of continuity, $f$ is not continuous.

Therefore, $f$ is not continuous at $x=1$.

(c)

To determine

Whether $f$ is continuous at $x=3$ by using the graph of $y=f\left(x\right)$. If it is not continuous then explain why not.

The graph of $y=f\left(x\right)$ is,

Answer to Problem 12CT

Solution:

$f$ is not continuous at $x=3$.

Explanation of Solution

Given information:

The graph of $y=f\left(x\right)$:

Explanation:

By observing the graph as $x$ approaches to $3$ from the left hand side the value of function approaches to $5$, as $x$ approaches to $3$ from the right hand side the value of function approaches to $-3$ and, $f\left(3\right)=-3$.

A function is continuous at $x=c$ when $\underset{x\to c^{-}}{\lim }f\left(x\right)=\underset{x\to c^{+}}{\lim }f\left(x\right)=f\left(c\right)$.

Here, $\underset{x\to 3^{-}}{\lim }f\left(x\right)\ne -3=\underset{x\to 3^{+}}{\lim }f\left(x\right)=f\left(3\right)$.

So, by using the definition of continuity, $f$ is not continuous.

Therefore, $f$ is not continuous at $x=3$.

(d)

To determine

Whether $f$ is continuous at $x=4$ by using the graph of $y=f\left(x\right)$. If it is not continuous then explain why not.

The graph of $y=f\left(x\right)$ is,

Answer to Problem 12CT

Solution:

$f$ is continuous at $x=4$.

Explanation of Solution

Given information:

The graph of $y=f\left(x\right)$:

Explanation:

By observing the graph as $x$ approaches $4$ from the left hand side the value of function approaches $-3.8$ and as $x$ approaches $4$ from the right hand side the value of function approaches $-3.8$ and at $x=4$ the value of function is $-3.8$.

A function is continuous at $x=c$ when $\underset{x\to c^{-}}{\lim }f\left(x\right)=\underset{x\to c^{+}}{\lim }f\left(x\right)=f\left(c\right)$.

Here, $\underset{x\to 4^{-}}{\lim }f\left(x\right)=\underset{x\to 4^{+}}{\lim }f\left(x\right)=f\left(4\right)=-3.8$.

So, by using the definition of continuity, $f$ is continuous.

Therefore, $f$ is continuous at $x=4$.