|Module 6 - Continuity|
|Introduction | Lesson 1 | Lesson 2 | Lesson 3 | Self-Test|
|Lesson 6.1: 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 "when" function 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. This 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-89, the connection between the formal and informal definitions may be more apparent.
Continuity at a Point
The formal definition says that a function f(x) is continuous at a point where x = c if
A function is continuous on an interval if it is continuous at every point in the interval.
The definition of continuity can be used to show that is continuous at x = .
From the Home Screen,
Because , is continuous at x = .
Discontinuity at a Point
If any of the three conditions in the definition of continuity fails at x = k, the function is discontinuous at that point.
6.1.1 Is continuous at x = 0? Justify your answer.
Click here for the answer.
You can visualize a discontinuity at a point by graphing the function in an appropriate window.
The function f(x) is already defined as .
The y-axis will need to be turned off in order to see the discontinuity at x = 0.
The discontinuity is represented as a hole whose coordinates are (0,1).
Removing the Discontinuity
The following shows how can be redefined so that it is continuous at x = 0.
Using the When Function
The "when" function on the TI-89 can be used to redefine f(x) to be for x 0 and 1 for x = 0. That is, .
To redefine f(x),
symbol is in the MATH Test submenu, which is found by pressing
The point of discontinuity has been removed, or in other words, the hole has been filled. Therefore, this redefined f(x) is continuous at x = 0.
Resetting the Axes
Before leaving this lesson you should turn the graphing axes back on.
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