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### Section 2.9 : Continuity

2. The graph of $$f\left( x \right)$$ is given below. Based on this graph determine where the function is discontinuous. Show Solution

Before starting the solution recall that in order for a function to be continuous at $$x = a$$ both $$f\left( a \right)$$ and $$\mathop {\lim }\limits_{x \to a} f\left( x \right)$$ must exist and we must have,

$\mathop {\lim }\limits_{x \to a} f\left( x \right) = f\left( a \right)$

Using this idea it should be fairly clear where the function is not continuous.

First notice that at $$x = - 8$$ we have,

$\mathop {\lim }\limits_{x \to - {8^ - }} f\left( x \right) = - 6 = \mathop {\lim }\limits_{x \to - {8^ + }} f\left( x \right)$

and therefore, we also know that $$\mathop {\lim }\limits_{x \to - 8} f\left( x \right) = - 6$$. We can also see that $$f\left( { - 8} \right) = - 3$$ and so we have,

$- 6 = \mathop {\lim }\limits_{x \to - 8} f\left( x \right) \ne f\left( { - 8} \right) = - 3$

Because the function and limit have different values we can conclude that $$f\left( x \right)$$ is discontinuous at $$x = - 8$$.

Next let’s take a look at $$x = - 2$$ we have,

$\mathop {\lim }\limits_{x \to - {2^ - }} f\left( x \right) = 3 \ne \infty = \mathop {\lim }\limits_{x \to - {2^ + }} f\left( x \right)$

and therefore, we also know that $$\mathop {\lim }\limits_{x \to \, - 2} f\left( x \right)$$ doesn’t exist. We can therefore conclude that $$f\left( x \right)$$ is discontinuous at $$x = - 2$$ because the limit does not exist.

Finally let’s take a look at $$x = 6$$. Here we can see we have,

$\mathop {\lim }\limits_{x \to {6^ - }} f\left( x \right) = 2 \ne 5 = \mathop {\lim }\limits_{x \to {6^ + }} f\left( x \right)$

and therefore, we also know that $\mathop {\lim }\limits_{x \to \,6} f\left( x \right)$ doesn’t exist. So, once again, because the limit does not exist, we can conclude that $$f\left( x \right)$$ is discontinuous at $$x = 6$$.

All other points on this graph will have both the function and limit exist and we’ll have $$\mathop {\lim }\limits_{x \to a} f\left( x \right) = f\left( a \right)$$ and so will be continuous.

In summary then the points of discontinuity for this graph are : $$x = - 8$$, $$x = - 2$$ and $$x = 6$$.