Inactive Tutor answered • 03/16/21
First check each function rule to make sure it is continuous. Second, check the boundaries between the pieces to see if they have the same function value.
Example:
Both f(x) = 4x + 1 and f(x) = (x + 1)2 are continuous by themselves. Now look at the boundary x = 2. Do both pieces have the same function value? In the case of 4x + 1 (since x cannot be 2), you can consider the limit as x approaches 2. That value is 9. Then (x + 1)2 when x = 2 is also 9. So the piecewise function is continuous.
Example:
Again, both pieces are continuous but at the boundary the value of the limit of 4x - 1 as x approaches 2 is 7. And the value of (x + 1)2 when x = 2 is 9. So this piecewise function is not continuous.
Example:
Despite the fact that the boundary has the same function values for both pieces (for x(4x + 1)/x the limit as x approaches 2 is 9 and the value of (x + 1)2 at x = 2 is 9), the individual piece x(4x + 1)/x is not continuous because the limit as x approaches zero does not exist. So this is piecewise function is not continuous.
