Ch.1 (Holt): CONTINUITY

I. Definition of Continuity

In this lesson you will explore continuity at a point, investigate discontinuity at a point, display discontinuities, and learn how to redefine a function to remove a point discontinuity. You will then use the TI-83 to graph piecewise defined functions.

Informally, a function is said to be continuous on an interval if you can sketch its graph on the interval without lifting your pencil off the paper. The formal definition of continuity starts by defining continuity at a point and then extends to continuity on an interval. The formal definition may not seem to have much in common with the concept of sketching a graph without lifting your pencil off the paper, but after investigating several examples with your TI-83, the connection between the formal and informal definitions should be more apparent. Continuity at a Point and on an Interval The formal definition of continuity at a point has three conditions that must be met. A function f(x) is continuous at a point where x = c if

1. exists 2. f(c) exists (That is, c is in the domain of f.)

3. A function is continuous on an interval if it is continuous at every point in the interval. Discontinuity at a Point The definition for continuity at a point may make more sense as you see it applied to functions with discontinuities. If any of the three conditions in the definition of continuity fails when x = c, the function is discontinuous at that point. Examine the continuity of when x = 0. Checking the Conditions for Continuity

1. The limit of the function as x approaches 0 was discussed in Lesson 6.3 where it

was shown that . Therefore, the first condition for continuity is satisfied.

2. The function is not defined at x = 0 because division by 0 is undefined. The function fails the second condition for continuity because 0 is not in the domain of f. 3. There is no need to verify the third condition because the second condition failed.

By the definition of continuity, you can conclude that is not continuous at x = 0.

1 Displaying Discontinuities A discontinuity at a point may be illustrated by graphing the function in an appropriate window. The discontinuity only shows up if it is at an x-value used in the plot. It is difficult (it might be impossible) to force it to show up at a point like or .

 Enter Y1 = sin(X)/X in the Y= editor.

The y-axis will need to be turned off in order to see the discontinuity at x = 0.

 Open the Format menu by pressing [FORMAT], which is above .

 Use the arrow keys to highlight "AxesOff" and press .

 Graph the function in a [-6, 6, 1] x [-1, 2, 1] window with Xres = 1.

 Explore the function with the Trace feature to see that is not defined at x = 0.

The discontinuity is represented as a hole in the graph at the point with coordinates (0,1).

Although the discontinuity appears as a hole in

2 the graph, it could be argued that no hole should appear because the missing point is infinitely small. On the TI-83 the missing point is represented by a missing pixel only if the x-value of the hole is an x-value used in the plot.

Removing the Discontinuity

The following shows how can be redefined to create a new function that is exactly like the original function for all non-zero values of x, but is continuous at x = 0.

Define a new function g(x) to be the function whose values are for and y = 1 for x = 0.

That is, This new function is called a piecewise function because different formulas are applied to different parts of the domain. The graph of g(x) is the same as the graph of except it includes the point (0,1), the point that fills the hole. Graphing a Piecewise Function

You can graph the piecewise function by entering the two pieces in Y1 and Y2. The first piece should already be in Y1. ( Y1 = sin(X)/X )

 Enter Y2 = 1/(X=0).

The "=" symbol can be pasted to the Y= editor from the Test menu by pressing [TEST] .

3  Display the new continuous graph by pressing .

The hole at (0, 1) has been filled. Turning the Axes Back On Before leaving this lesson you should turn the graphing axes on.

 Return to the Format menu by pressing [FORMAT].  Use the arrow keys to highlight "AxesOn."

 Select this setting by pressing .

 Exit the Format menu by pressing [QUIT].

Practice Exercise: Redefine to make it continuous at x = 2. Click here for the answer.

II. 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.  Set Xres = 1.

 Enter the expression Y1 = abs(X)/X in the Y= Editor.

The "abs(" function can be pasted from the catalog by pressing [CATALOG] .

4  Graph the function in a ZDecimal window.

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 any jump discontinuity, the one-sided 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 y-axis.

 Graph in a [-2, 2, 1] x [-2, 2, 1] window.

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).

5  Display the Zoom menu by pressing .

 Select the Zoom Box (1:ZBox) feature by pressing .

This feature will allow you to magnify the region around the y-axis by drawing a box around the part of the window you want to magnify.  Move the cursor to the upper left corner of the desired box by using the arrow keys.

 Mark the upper left corner by pressing .  Move the cursor to the diagonally opposite corner by using the arrow keys. You should see the outline of a box evolve as you move the cursor.

6  Finish identifying the box by pressing 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.  Use the Zoom Box feature to zoom in further around the y-axis.

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 right-hand limits do not exist in this case.

ZBox Feature The first corner that you mark when you use zoom box can be any one of the four corners, not just the upper left corner as in the example above. The only requirement is that the second corner be diagonally opposite the first.

An Infinite Discontinuity

The function y = tan(x) has a third type of discontinuity at . This discontinuity is called an infinite

7 discontinuity.

 Graph y = tan(x) in a window.

Practice Exercise: Based on the graph, why do you think y = tan(x) is said to have an infinite discontinuity at ?

III. Piecewise Functions

In this lesson you will modify a piecewise function to make it continuous and then graph the function using Boolean operators.

Consider the function defined by Practice Exercise: By using the definition of continuity, find the value of k that makes the function continuous at x = 2. Graphing a Piecewise Function

Display the graph of using the value of k that makes the function continuous.  Set Xres = 1.

 Enter

You can paste " " from the Test menu by pressing [TEST] . [TEST] is above .

2  Enter Y2 = ( (1/2)X +3 ) / (X>2) The symbol ">" is in the TEST menu.  Graph the function in a [-3, 7, 1] x [-3, 25, 5] window.

8 The graph appears to be continuous at x = 2. Practice Exercise: Describe how the graph would change if you changed k to 7. Illustrate the result of changing the value of k to 7.

2  Set Y2=( (1/2)X +7 ) / (X>2).

 Display the graph of the function by pressing .

The second part of the piecewise function has been shifted upward, and the function is no longer continuous at x = 2. The discontinuity in this modified function (k = 7) is a jump discontinuity. Practice Exercise: Evaluate the left- and right-hand limits of the piecewise function with k = 7 at x = 2.

Self Test

1. Which of the following functions are continuous at the given point?

I.

9 II.

III.

2. What type of discontinuity does have at x = -1? Illustrate the discontinuity with a graph.

3. Let . For what value of k will f(x) be continuous at x = 1?

4. Graph using Boolean expressions.

5. Redefine so that it is continuous at x = 0.

Self Test

Answer 1

II and III

Answer 2

A removable discontinuity (a hole) exists and the hole can be seen in a zoom decimal window at

.

10

Answer 3 k = 0

Answer 4

Answer 5

11