Chapter 2.4, Problem 15E

To determine

To explain:the function isdiscontinuous at the given number $a$ . Sketch the graph of the function.

Explanation of Solution

Given:

  $f\left(x\right)=\{\begin{array}{l}{e}^{x}ifx<0\\ x^{2}ifx\ge 0\end{array}a=0$ .

Calculation:

The given function is $f\left(x\right)=\{\begin{array}{l}{e}^{x}ifx<0\\ x^{2}ifx\ge 0\end{array}$ .

Now, sketch the graph of the given function.

The graph of the functionis:

  

From the above graph, it is clearly seen that:

  $\underset{x\to 0^{-}}{\lim }f\left(x\right)=1$

And $\underset{x\to 0^{+}}{\lim }f\left(x\right)=0$

Therefore, $\underset{x\to 0}{\lim }f\left(x\right)$ does not exist.

Which means $f\left(x\right)$ is not continuous at $x=0$ .

Hence, this type of discontinuity is known as jump discontinuity.