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 access the built-in example functions, use the pull-down menu in the applet.

The functionality of the applet also allows for user input of multiple parameter (6, in this applet) piecewise functions. The syntax for entering piecewise functions is illustrated using the following examples.

The values of all parameters are whole number multiples of -0.25 and +0.25.

The syntax will probably not be transparent for most students so use careful judgment in whether or not to have students do coding of additional 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.

If you provide the applet for student assignments/experimentation, please point out what the appropriate parameters are for the particular example.  These can be observed from the function coding, however, they are also noted in the table below.  If a parameter is NOT relevant in a particular example, the slider will have no effect.

The built-in functions are given below.

Example 1: One parameter

NOTES/Parameter List

Parameter is k

 

Example 2: One parameter

Parameter is k

Example 3:  One parameter

Parameter is k
Example 4:  Two parameters
Parameters are m and b
Example 5:  Two parameters

Parameters are m and b
Example 6:  Two parameters

 

Parameters are a and b

Example 7:  One parameter

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 and the point at (x,k) is illustrated using a very short line segment.  The object is to "plug the hole" in the graph using the slider.
Watch carefully--the short line segment is very hard to see.

Parameter is k

Example 8: Two parameters

 


HINT:  Set ymax to 12

Parameters are a and b

Example 9:  Two parameters

As in Example 7, the point at (2,5) is illustrated using a very short line segment because the applet does not show points. Watch carefully--the short line segment is very hard to see.

Parameters are a and b

Example 10: Four parameters

As in Examples 7 and 9, the point at (0,4) is illustrated using a very short line segment. Watch carefully--the short line segment is very hard to see.

Parameters are a, b, c, and d

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

last updated 11/03/2004  DRH