Module 8  Continuity  
Introduction  Lesson 1  Lesson 2  Lesson 3  Self Test  
Lesson 8.2: Types of Discontinuities  
The discontinuity you investigated in Lesson 8.1 is called a removable discontinuity because it can be removed by redefining the function to fill a hole in the graph. In this lesson you will examine three other types of discontinuities: jump, oscillating, and infinite. A Jump Discontinuity The function has a jump discontinuity at x = 0. Graphing the function can illustrate the reason for the name of this type of discontinuity.
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 righthand limits are defined as x approaches 0, but is undefined because For any jump discontinuity, the onesided limits are unequal. An Oscillating Discontinuity The function has a discontinuity at x = 0 because it is not defined at x = 0. This discontinuity is called an oscillating discontinuity. The oscillating discontinuity can be illustrated by viewing its graph in several windows that increasingly zoom in around the yaxis.
It is hard to tell what is happening near x = 0 by looking at the graph. You can magnify a portion of the graph near x = 0 by using a feature in the Zoom menu called ZBox (Zoom Box).
This feature will allow you to magnify the region around the yaxis by drawing a box around the part of the window you want to magnify.
You should see the outline of a box evolve as you move the cursor.
The region 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, the graph will continue to oscillate between y = 1 and y = 1. In fact, there are infinitely many oscillations packed near x = 0. That is why is said to have an oscillating discontinuity at x = 0. The left and righthand limits do not exist in this case.
An Infinite Discontinuity The function y = tan(x) has a third type of discontinuity at . This discontinuity is called an infinite discontinuity.
8.2.1 Based on the graph, why do you think y = tan(x) is said to have an infinite discontinuity at ? Click here for the answer. 

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