Question

For the following exercises, consider the function shown in Figure 17.

What condition of continuity is violated at these points?

Answer 1

A function f (x) is continuous at x = a only if it satisfies all the three following conditions:

• Condition 1: f (a) exists.

• Condition 2: limx→af(x) exists

• Condition 3: limx→af(x) = f(a)

 

Consider the graph provided in the exercise:

 

At x = -1, limx→-1-f(x) ≠ limx→-1+f(x)

So, limx→-1f(x) does not exist.

 

So, condition 2 fails.

At x = 1, f(1) does not exist.

 

So condition 1 fails.

At x = 2, limx→-2-f(x) ≠ limx→2+f(x)

 

So, limx→2f(x) does not exist.

So condition 2 fails.

Condition 2 fails.