Question

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

Is f continuous at 2?

Short answer

The answer is no. The function is not continuous at 2.

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 $f\left(2\right)$which is equal to 3 as given in the graph.

Now, we calculate LHL which is,

$\underset{x\to 2^{-}}{\lim }f\left(x\right)=1$.

Finally we see RHL which is,

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

From here we can see that $LHL\ne RHL\ne f\left(2\right)$

So, the conditions to check continuity fails. That's why, the function is not continuous at $x=2$.

And from the graph itself we can clearly see that we have to pick our pencil up the paper when we move through $x=2$which is also a method to check whether the function is continuous or not.