Question

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

Is f continuous at -4?

Short answer

The answer is no. The function is not continuous at -4.

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 $f(-4)$which is equal to 2 as given in the graph.

Now, we calculate LHL which is,

$\underset{x\to -4^{-}}{\lim }f\left(x\right)=2$.

Finally we see RHL which is,

$\underset{x\to -4^{+}}{\lim }f\left(x\right)=-2$.

From here we can see that $LHL\ne RHL$

So, the condition to check continuity fails. Thatswhy, the function is not continuous at $x=-4$.

And from the graph itself we can clearly see that we have to pick our pencil up the paper when we move through$x=-4$ which is also a method to check whether the function is continuous or not.