Investigating Continuity of Piecewise Functions

This is a collection of examples suitable for in-class demonstration and/or individual student exploration.

The examples are built into a Java applet from the Java Components for Mathematics collection.   To launch the applet, press the button "Launch MultiGraph" below the example listing.  To access the built-in example functions, use the pull-down menu in the applet and then press Load Example.

The functionality of the applet also allows for user input of one or two parameter piecewise functions. The syntax for entering piecewise functions is illustrated using the following examples.

The function

would be entered as

(x<-2)?3*x+2:x-1.

The function

would be entered as

(x<-2)?3*x+2:(-2<=x AND x < 1)?2*x^2-x+1:(x>=1)?x-1.

Note that axis limits can be adjusted as necessary.

Endpoints are not displayed by the Java applet so it is important to discuss endpoint inclusion/exclusion as appropriate.

The built-in functions are given below.

 Example 1: One parameter NOTES/Analytic Solution k = 1 Example 2: One parameter HINT:  Set ymax to 10 k = 1/2 Example 3:  One parameter k = 4 Example 4:  Two parameters k = -1 m = 1 Example 5:  Two parameters HINT:  Set ymax to 18 k = -3 m = 4 Example 6: NOTE:  The function is not continuous at x = k because it is undefined at the point where x = k.  However, the limit as x approaches k of the function does exist because , provided x is not equal to k.  Thus, the graph of g is a line that has a "hole" when x = k. Because this applet does not show endpoints, the graph is rigged to have a break.  The object is to "plug the hole" in the graph using the slider.

The Java applet is part of the Java Components for Math collection, developed by David Eck (eck@hws.edu).This project was supported by NSF grant number DUE-9950473