Example   Given the graph of f(x), illustrated below, find out if f(x) is continuous at x = -2 , x = 0 , and x = 3 .

Solution

To give answer of the question for each point we'll have to get both the limit at that point and the function value at that point. If they are equivalent the function is continuous at that point and if they aren't the function isn't continuous at that point.

First

x = -2 .                    

                                  f ( -2) = 2          doesn't exist

The function value & the limit aren't the similar and hence the function is not continuous at this point. This sort of discontinuity in a graph is called a jump discontinuity. Jump discontinuities takes place where the graph has a break in it as this graph does.

Now x = 0 .

                    f (0) = 1     

The function is continuous at this point as the function & limit have the similar value.

At last x = 3 .

f (3) = -1       

At this point the function is not continuous. This kind of discontinuity is called as a removable discontinuity.Removable discontinuities are those there is a hole within the graph as there is in this case.