Chapter 2.3, Problem 15E

To determine

Tofind:The value that can be assigned to $f\left(2\right)$ such that the extended function is continuous at $x=2$ .

Answer to Problem 15E

The value 0 can be assigned to $f\left(2\right)$ such that the extended function is continuous at $x=2$ .

Explanation of Solution

Given information:

The given function is $f\left(x\right)=\{\begin{array}{l}{x}^2-1,-1\le x<0\\ 2x,0<x<1\\ 1,x=1\\ -2x+4,1<x<2\\ 0,2<x<3\end{array}$ and the graph of the function is given below:

  

Calculation:

As observed from the graph, the left hand and right hand limits of function $f\left(x\right)$ at $x=2$ is equal to $0$ .

  $\underset{x\to 2}{\lim }f\left(x\right)=0$

According to the definition of continuity of $f\left(x\right)$ at $x=a$ the must condition is $\underset{x\to a}{\lim }f\left(x\right)=f\left(a\right)$ .

For the continuity of $f\left(x\right)$ at $x=2$ , the condition is $\underset{x\to 2}{\lim }f\left(x\right)=f\left(2\right)$ and $\underset{x\to 2}{\lim }f\left(x\right)$ is equal to $0$ .So, the value of $f\left(2\right)$ must be $0$ for $f\left(x\right)$ to be continuous at $x=2$ .

Therefore, the value 0 can be assigned to $f\left(2\right)$ such that the extended function is continuous at $x=2$ .