|Module 8 - Continuity|
|Introduction | Lesson 1 | Lesson 2 | Lesson 3 | Self Test|
|Lesson 8.1: 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 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
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
By the definition of continuity, you can conclude that is not continuous at x = 0.
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 .
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 in the graph at the point with coordinates (0,1).
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.
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 )
The hole at (0, 1) has been filled.
Turning the Axes Back On
Before leaving this lesson you should turn the graphing axes on.
8.1.1 Redefine to make it continuous at x = 2. Click here for the answer.
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