Intermediate Value Therem

A function that is continuous on an interval has no gaps and hence cannot "skip over" values. If a function is continuous on a closed interval from x = a to x = b, then it has an output value for each x between a and b. In fact, it takes on all the output values between f (a) and f (b); it cannot skip any of them. More formally, the Intermediate Value Theorem says:

Let f be a continuous function on a closed interval [a,b]. If k is a number between f (a) and f (b), then there exists at least one number c in [a,b] such that f (c) = k.

The following applet will help understand what this means. We will look at the interval [0,2] for several functions.

=0 & x<= 2?x^2;-2,4,-2,4,0;yinput,1;closed,0,0;closed,2,4"> =0 & x<= 2?2sin(2x);-2,4,-2,4,0;yinput,1;closed,0,0;closed,2,-1.513605"> =0 & x<= 2?1/(x-1);-2,4,-2,4,0;yinput,0.5;asymptote,1,0,1,1;closed,0,-1;closed,2,1"> =0 & x<= 2?(x<1?x:x+1);-2,4,-2,4,0;yinput,1.5;open,1,1;closed,1,2;closed,0,0;closed,2,3"> =0 & x<= 2?x(x-1)/(x-1);-2,4,-2,4,0;yinput,1;open,1,1;closed,0,0;closed,2,2">

Try the following:

1. The first graph shown, a piece of a parabola, is continuous on [0,2]. If k = 1, is there some input value of c that will make f (c) = k ? Move the c slider, or type a guess into the input box for c, so that the crosshair is horizontally at the same level as y = k = 1. Of course, the answer is that c = 1. Note that for any k from 0 to 4 (which are just the values of f (0) and f (2) ) there is some c that will give you this value out of the function.
   
   
2. Select the second example. This piece of a stretched sine curve is also continuous on [0,2]. If k = 1, is there some input value of c that will make f (c) = k ? Move the c slider, or type a guess into the input box for c to find a c that makes y = k = 1. In this case, there are two possible answers for c that work. This is okay, as the Intermediate Value Theorem only says that there will be at least one c, not that there will be exactly one.
   
   
3. Select the third example. This function has a vertical asymptote at x = 1 and so is not continuous. If k = 0.5, is there some input value of c that will make f (c) = k ? Move the c slider, or type a guess into the input box for c . In this case nothing works. The discontinuity allows the function to "skip over" y = 0.5, and in fact skips over all the output values between -1 and 1.
   
   
4. Select the fourth example. This function jumps from 1 to 2 at x = 1, called a jump discontinuity and so is not continuous. If k = 1.5, is there some input value of c that will make f (c) = k ? Move the c slider, or type a guess into the input box for c. In this case nothing works. The discontinuity allows the function to "skip over" y = 1.5.
   
   
5. Select the fifth example. This function has a hole in it at x = 1, called a removable discontinuity and so is not continuous. If k = 1, is there some input value of c that will make f (c) = k ? Move the c slider, or type a guess into the input box for c . In this case nothing works. The discontinuity allows the function to "skip over" y = 1.

As you have just seen, the Intermediate Value Theorem only holds for continuous functions. Discontinuities allow a function to "skip over" values.

For more information on rights and downloading, refer to http://www.calculusapplets.com/download.html.

© Copyright 2001 David J. Eck

© Copyright 2007 Thomas S. Downey

Permission is hereby granted, free of charge, to any person obtaining a copy of this software and associated documentation files (the “Software”), to deal in the Software without restriction, including without limitation the rights to use, copy, modify, merge, publish, distribute, sublicense, and/or sell copies of the Software, and to permit persons to whom the Software is furnished to do so, subject to the following conditions:

The above copyright notice and this permission notice shall be included in all copies or substantial portions of the Software.

You must cause any files that you modify to carry prominent notices stating that you changed the files and the date of any change, and modified files must be put into a Java package different from edu.hws.

THE SOFTWARE IS PROVIDED “AS IS”, WITHOUT WARRANTY OF ANY KIND, EXPRESS OR IMPLIED, INCLUDING BUT NOT LIMITED TO THE WARRANTIES OF MERCHANTABILITY, FITNESS FOR A PARTICULAR PURPOSE AND NONINFRINGEMENT. IN NO EVENT SHALL THE AUTHORS OR COPYRIGHT HOLDERS BE LIABLE FOR ANY CLAIM, DAMAGES OR OTHER LIABILITY, WHETHER IN AN ACTION OF CONTRACT, TORT OR OTHERWISE, ARISING FROM, OUT OF OR IN CONNECTION WITH THE SOFTWARE OR THE USE OR OTHER DEALINGS IN THE SOFTWARE.