You should use one of the computer algebra systems below with this module. Click on the appropriate icon for your preferred CAS and then arrange your screen so that you can easily move back-and-forth between this window and your CAS window. Click on the appropriate help button for help.
The first three of these functions are continuous. If you make a small change in the input then the result is a small change in the output -- for example, as you turn the knob on a faucet the waterflow increases steadily without any sudden jumps.
The last function, however, is different. The postage tables typically look something like the following.
Weight Postage
less than 1 pound $0.75
1 pound - 2 pounds $1.25
2 pounds - 3 pounds $1.65
Notice there are two sudden jumps. If the weight of a package is the least little bit less than 1 pound then the postage is only $0.75 but as soon as the weight reaches 1 pound, the price jumps to $1.25. There is a similar jump at 2 pounds.
Functions like the postage function above are hard to deal with, mathematcally and practically. The author once brought a package into the post office and the clerk announced that because its weight was just above 1 pound the price would be very high. He also noticed that I had put a tremendous amount of tape and string on the package and suggested I repack it with less string. I cut some string off but that didn't do it. He made me rewrap it using less tape but it still was just a wee bit above 1 pound. When he started quizzing me about how much packing material I had put in and suggesting that I go home and return tomorrow after completely repacking it, I gave up and said it wasn't worth all that work. He gave me a very dirty look and charged me the higher price.
The source of my post office problem was that just a tiny change in the weight of the package caused a substantial change in the postage. This kind of nasty behavior in a function is called a discontinuity.
Because a function may be discontinuous (nasty) at some points and continuous (nice and well-behaved) at other points we discuss these ideas point-by-point.
Definition:
A function y = f(x) is said to be continuous at the point a if
Lim f(x) = f(a)
x --> a
Like many important ideas the idea of continuity can be expressed in several different ways. Here is another way of expressing exactly the same idea
A function f(x) is continuous at the point a if it is defined at the point a and whenever x is close to a then f(x) is close to f(a).The same ideas that are involved in estimation and limits are involved in continuity. In practical situations we need to be more precise about the two uses of the word close.
A function f(x) is said to be continuous at the point a if it is defined at a and for any permissible error epsilon for the output there is an input tolerance delta such that if |x - a| < delta then |f(x) - f(a)| < epsilon.
A function is said to be continuous if it is continuous at every point.
Next we look at a movie that shows one way of visualizing a function y = f(x). You can look at this movie in your CAS window or by clicking on the icon below.
This particular movie looks at the function y = x^2. The vertical line on the left represents the input to this function -- symbolized by the variable x. The vertical line on the right represents the output of this function -- symbolized by the variable y. We can draw a line from each input on the left to the corresponding output on the right. For example, in the first frame of this movie there is a blue line going from the input -1 to the output 1 and a red line going from the input 0.5 to the output 0.25. This first frame shows graphically the facts that
f(-1) = 1 and
f(0.5) = 0.25
As the movie plays or as you slide the slider bar at the bottom of the movie window, the blue input on the left changes and the blue line automatically goes to the corresponding output on the right. The red input and output stay fixed for comparison. Because this particular function is continuous, as the input changes the output changes without any sudden abrupt jumps.
Our next movie available in either your CAS window or by clicking on the icon below looks at a function with a discontinuity. Play that movie and compare its behavior with the behavior shown in the first movie. By modifying your CAS window you can look at other examples.
Continuous functions have lots of nice properties. One of these properties is that the graph of a continuous function can be drawn using one continuous line without lifting the pencil from the paper. Computer graphics programs take advantage of this property. They actually compute only a few points on a graph and then connect those points by straight lines. The human eye usually doesn't notice the difference between the true curve and a computer drawn curve made up of many short straight line segments. Unfortunately, however, when the graph of a function with discontinuities is drawn in this way it is frequently misleading. For example, here is what happens when we ask Mathematica to graph the function.
The true graph looks like the figure below.
Notice there is a break in the true graph but not in the Mathematica graph. In the true graph we have very carefully used a solid dot at the point (1, 2) because this point is on the graph and we have placed a hollow dot at the point (1, 1) because this point is not on the graph. The graph drawn by Mathematica shows a line going from the point (1, 1) to the point (1, 2). This line is spurious.
Most modern CAS systems work very hard to avoid this problem. Some of the questions below ask you to see how your CAS system handles graphs of functions with discontnuities.
A continuous function