Question

In Problems 13–32, use the accompanying graph of y = f(x).

Is f continuous at 0?

Short answer

The answer is Yes. The function is continuous at 0.

Step-by-step solution

Step. 1 How to check continuity? 

For a function to be continuous its LHL = RHL = f(c) at the point $x=c$.

Step. 2 Calculating LHL, RHL and f(c). 

Firstly we calculate $f\left(0\right)$which is equal to 3 as given in the graph.

Now, we calculate LHL which is,

$\underset{x\to 0^{-}}{\lim }f\left(x\right)=3$.

Finally we see RHL which is,

$\underset{x\to 0^{+}}{\lim }f\left(x\right)=3$.

From here we can see that $$\begin{gathered} LHL=RHL=f\left(0\right) \\ \underset{x\to 0^{-}}{\lim }f\left(x\right)=\underset{x\to 0^{+}}{\lim }f\left(x\right)=f\left(0\right)=3 \end{gathered}$$

So, all the conditions to check continuity passes. That's why, the function is continuous at $x=0$.

And from the graph itself we can clearly see that we don't have to pick our pencil up the paper when we move through $x=0$which is also a method to check whether the function is continuous or not.