4.3B   Is Piecewise Functions Continuous?

Is a Piecewise Function Continuous?

 It is easy to see graphically if a piecewise function is continuous.

 Most of the time, you are given the piecewise function without its graph and are asked to determine its continuity.

 How do you check for the continuity of a piecewise function without a grapher or Desmos?  The answer is below...

Note: At the right is an activity to complete.

Algebraically 

Recalling how to determine if a function is continuous, we will review the definition of continuity...

The 3 questions you should be asking yourself...    Does the limit exist?

  Test 1: Is the limit of f(x) as x approaches a from the left, equal to the limit of f(x) as x approaches a from the right?

             If they are equal, then the limit exists.

             If not, then do not test for the existence of f(a).

Test 2: Does the value of f(a) exist? 

                      Plug in an "a" in the function, f(x).  Find f(a).

Test 3: Does the lim f(x) = f(a)? 

               If yes, then the function is continuous.

                        If no, then the function is discontinuous.

***Anytime you are asked to determine if a function is continuous, you MUST use limit notation and test the 3 above conditions.***