Chapter 1, Problem 21RE

To determine

To find: Whether the function $f\left(x\right)$ shown in figure below is continuous at point $x=a$ .

  

Answer to Problem 21RE

The function $f\left(x\right)$ is continuous at $x=a$ .

Explanation of Solution

Given:

The graph for the function $f\left(x\right)$ is shown below.

  

Concept used:

The graph of the function $g\left(x\right)$ is continuous at some point $x=a$ in domain of $g\left(x\right)$ , if $\underset{x\to a}{\lim }g\left(x\right)=g\left(a\right)$ .

Calculation:

From the function $f\left(x\right)$ , it can be interpreted that the domain of the function $f\left(x\right)$ is real. And, it can be interpreted from the graph that as $x\to a^{-}$ (from left), the value of $\underset{x\to a^{-}}{\lim }f\left(x\right)$ is defined and finite because of dark hole at $x=a$ . Similarly, when $x\to a^{+}$ (from right) the value of $\underset{x\to a^{+}}{\lim }f\left(x\right)$ is again defined and finite because of dark hole at $x=a$ .

Also, it can also be interpreted from the graph that $\underset{x\to a}{\lim }f\left(x\right)=f\left(a\right)$ because of dark hole at $x=a$ .

Conclusion:

Hence, the function $f\left(x\right)$ is continuous at $x=a$ .