William W. answered • 03/16/21
Experienced Tutor and Retired Engineer
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.
