|Module 6 - Continuity|
|Introduction | Lesson 1 | Lesson 2 | Lesson 3 | Self-Test|
|Lesson 6.2: Types of Discontinuities|
The discontinuity you investigated in Lesson 6.1 is called a removable discontinuity because the discontinuity can be removed by redefining the function in order to fill a hole in the graph. In this lesson you will examine three other types of discontinuities: jump, oscillating and infinite.
The function has a jump discontinuity at x = 0. Graphing can illustrate the reason for the name of this type of discontinuity. Be sure to set xres = 1.
The "abs(" function can be pasted from the catalog by pressing .
The function is not continuous at x = 0 because x = 0 is not in the domain of the function. The function is said to have a jump discontinuity because it jumps from y = -1 to y = 1 at x = 0. The left- and right-hand limits are defined as x approaches 0, but is undefined because . For a jump discontinuity, the one-sided limits are unequal.
Because is in the Y= Editor, y1 may be used in place of the expression in the limit function.
y1(x) can be typed directly into the Edit Line by pressing .
The function has a discontinuity at x = 0 because it is not defined at x = 0. It also has an oscillating discontinuity at x = 0. A few graphs will illustrate this.
It is hard to tell what is happening near x = 0. You can use a feature in the Zoom menu called ZoomBox to magnify a portion of the graph near x = 0.
This feature will allow you to draw a box around the part of the graph you would like to magnify.
When the TI-89 prompts you for the first corner of the box,
The cursor changes shape and the calculator prompts you for the second corner of the box, which should be diagonally opposite the first corner.
You should see the outline of a box evolve as you move the cursor.
The area inside the box is magnified to fill the entire viewing screen.
The Zoom Box feature magnifies the graph by adjusting the Window values to match the box corner coordinates.
This window shows even more oscillations than the previous one. No matter how many times you zoom in around the y-axis, the graph will continue to oscillate wildly. In fact, there are infinitely many oscillations packed near the y-axis. That is why is said to have an oscillating discontinuity at x = 0. Notice that the left- and right-hand limits do not exist in this case.
The function y = tan(x) has an infinite discontinuity at .
6.2.1 Based on the graph, why do you think y = tan(x) is said to have an infinite discontinuity at ? Evaluate the left- and right-hand limits at . Click here for the answer.
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