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:
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.
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.
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.
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.
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.