For the following exercises, consider the function shown in Figure 17. What condition of continuity is violated at these points?

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

What condition of continuity is violated at these points?

Solutions

Expert Solution

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.

Junaid answered 1 year ago

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