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