Here we see a consequence of a function being continuous.
If you are more of a visual person, you should imagine a continuous function, where you know the value of the function at two endpoints, and , but you don’t really know what the function does between the points and :
Now, let’s contrast this with a time when the conclusion of the Intermediate Value Theorem does not hold.
Building on the question above, it is not difficult to see that each of the hypothesis of the Intermediate Value Theorem is necessary.
Let’s see the Intermediate Value Theorem in action.
This example also points the way to a simple method for approximating roots.
The Intermediate Value Theorem can be used to show that curves cross:
To start, note that both and are continuous functions on the interval , and hence is also a continuous function on the interval . Now
and in a similar fashion
Since and we see that the expression above is positive. Therefore, , and by the Intermediate Value Theorem, there exist a number in such that But this means that and that . Therefore, the curves and intersect at the point .
We can see this point of intersection by looking at the graphs of and on the given interval.
- They start and finish drinking at the same times.
- Roxy starts with more water than Yuri, and leaves less water left in her bowl than Yuri.
- the amount of water in Roxy’s bowl at time .
- the amount of water in Yuri’s bowl at time .
Now if is the time the cats start drinking and is the time the cats finish drinking. Then we have and Since the amount of water in a bowl at time is a continuous function, as water is “lapped” up in continuous amounts, is a continuous function, and hence the Intermediate Value Theorem applies. Since is positive when at and negative at , there is some time when the value is zero, meaning meaning there is the same amount of water in each of their bowls.
And finally, an example when the Intermediate Value Theorem does not apply.
- They start and finish eating at the same times.
- Roxy starts with more food than Yuri, and leaves less food uneaten than Yuri.
- the amount of dry cat food in Roxy’s bowl at time .
- the amount of dry cat food in Yuri’s bowl at time .
However in this case, the amount of food in a bowl at time is not a continuous function! This is because dry cat food consists of discrete kibbles, and is not eaten in a continuous fashion. Hence the Intermediate Value Theorem does not apply, and we can make no definitive statements concerning the question above.
For some interesting extra reading check out: