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.

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See About the calculus applets for operating instructions.

1. A continuous function

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. Also continuous

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.

This example shows how the Intermediate Value Theorem only ensures output values between f(a) and f(b) even though there are more values outside this part of the range. We can't tell without looking at the graph that there are other output values but the theorem guarantees it between a and b without having to look at the graph.

3. Essential discontinuity

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. Jump discontinuity

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. Removable discontinuity