Piecewise Functions:  Investigating Continuity

Objective:  Students in calculus need to be proficient in working with functions in a variety of ways:  graphical, numerical, analytic, or verbal, and to understand connections among these representations.  This demo provides a set of visualizations designed to help students better understand what it means for a piecewise function to be continuous at a particular domain value.

Level:  The visualizations and activities in this demo are appropriate for high school or college level calculus classes.  Although the examples specifically discuss continuity, the verbal discussion can be easily changed to be appropriate for a precalculus class.

Platform:  The movies in the demo require Quicktime player, however, additional platforms are described and where appropriate, accompanying files and/or links are provided.  The Quicktime movies may be downloaded by right clicking and saving the movie to your computer.  Excel files that accompany this demo are also freely downloadable and can be used in class or for individual investigations by students.  The Java applets use the Java Components for Math, developed by David Eck under NSF grant number DUE-9950473.

Instructor's Notes:  It is often the case that my students try to memorize procedures instead of striving to gain an understanding of fundamental mathematical concepts.  It has been my experience that students often have great difficulty understanding piecewise functions and when these functions are continuous at a "breakpoint" in the domain.  The visualizations in this demo were developed with the idea that if a student gains geometric insight, then an analytic approach will become more meaningful.

A recent study [1] suggests that in science classes, a discussion prior to executing the demo and a post-discussion where students articulate and write down what they observed can be beneficial to student learning.  Although the study focused on undergraduate physics students, this approach makes sense in mathematics classes as well.

A separate, similarly constructed demo has been developed to investigate differentiability of piecewise functions.  It can be found by clicking here

Example 1A One Parameter Problem

A common problem in precalculus and calculus is to determine a value of a parameter for which a given piecewise function is continuous.  For example, suppose we wish to determine the value of k for which the piecewise function

is continuous at x = 3.  As instructors, we know that the analytic approach involves setting the two "pieces" equal when x = 3 and solving for k.  However, students often do not realize that by equating the "pieces," we are really requiring that the left and right portions of the graph match up (have the same y value) at x = 3.

Intuitively, we want students to understand that continuity of a piecewise function requires that the pieces of the graph fit together (or match) at the domain value in question.  After students understand the graphical interpretation of the problem, they can then begin to understand the analytic approach to solving the problem, rather than simply trying to memorize a procedure.

The following movie provides compelling evidence that there (probably) is such a value of k (for this example).   Before playing the movie, the following questions may help your students to understand what they will see in the animation.

1. Which part of the graph (left or right) is changed when k is changed?  How does the graph change?

2. Do you think there is a value of k for which the left and right pieces of the graph will "match" at x = 3?

3. If there is such a value of k, describe what must be true about the value of the left and right hand function expressions at x = 3.

After students have discussed what they think will happen, play the video.

 The left piece of the function changes as k changes.  The left piece of the graph is in red. Click the PLAY button to initiate the movie.

The Quicktime movie provides start/stop features so that the instructor can pause during the course of the lesson to discuss what is happening in the animation.  The functionality of the movie control bar is shown below.

After playing the video, ask your students to discuss what they observed.  The third question above is designed to make the connection between the graphical concepts and the analytical procedure to obtain the exact value of k that is required for the function to be continuous at x = 3.

The analytical counterpart to having the pieces of the graph match at x = 3 is that the left and right hand limits of the function as x approaches 3 must be equal.  For this example, this requirement is equivalent to solving the following equation for k.

Thus, the function is continuous at x = 3 when k = 1.

A Java implementation of Example 1 is shown below.  The object is to use the sliders to  "match" the pieces of the graph so that the function is continuous.

The applet does not provide for the functionality of including/excluding endpoints so in the class discussion it is important  that endpoint behavior is explored.

Example 2:  A Two Parameter Problem

We wish to find the values of a and b so that the function with equation

is continuous for all x. This involves finding values of the parameter so that f is continuous at x = 1 and x = 3.

The following movie provides compelling evidence that there (probably) are such values of a and b.   Before playing the movie, the following questions may help your students to understand what they will see in the animation.

1. Which part(s) of the graph (left, middle, right) is(are) changed when a and b are changed?  How does the graph change?

2. Do you think there are values of a and b for which the middle piece of the graph will match the left and right pieces at x = 1 and x = 3, respectively?

3. If there are such values, describe what must be true about the values of the left, middle, and right hand function expressions at x = 1 and x = 3.

 The left and right pieces correspond to the parabolic parts of the function definition (blue); the linear middle section is in red. Click the PLAY button to initiate the movie.

To analytically solve the problem involves evaluating left and right hand limits at x = 1 and at x = 3.  This leads to two equations:

Simplifying, we obtain the system of equations

which has solution a = -3 , b = 4.

A Java implementation for this example is shown below.  As in the previous example, the object is to use the sliders to make the pieces of the graphs match at x = 1 and x = 3.

A collection of examples has been constructed that can be used for classroom demonstration as well as for individual student exploration.

• An interactive Java applet is available that has several built-in examples plus the functionality of entering additional examples (PLEASE READ IMPORTANT INFORMATION ABOUT SYNTAX BEFORE DECIDING WHETHER TO LET STUDENTS INPUT THEIR OWN EXAMPLES).  That applet can be accessed by clicking on this link.

• An Excel file containing ten interactive examples can be downloaded from here. NOTE:  In order to use the control features in the file, you must choose "Enable Macros."

• A collection of Quicktime movies to illustrate one-parameter problems can be downloaded from here.

References

1.  Crouch, Catherine H., Fagen, Adam P., Callan, J. Paul, and Mazur, Eric. "Classroom Demonstrations:  Learning tools or entertainment?" Am. J. Physics, 72(6), June 2004, pp 835-838.

Credits:  This demo and the complementary demo on differentiability of piecewise functions, were inspired by free response questions on the 2003 AP Calculus AB Exam (#6) and by questions submitted by students to Ask Dr. Math at Mathforum@Drexel. The Java applets were configured from Java Components for Math,  developed by David Eck under NSF grant number DUE-9950473.  Excel files were developed by David R. Hill at Temple University and Lila F. Roberts at Clayton State University.

LFR 8/14/04  Last updated 9/15/2010 DRH

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