Here we see a consequence of a function being continuous.

The Intermediate Value Theorem should not be brushed off lightly. Once it is understood, it may seem “obvious,” but mathematicians should not underestimate its power.
PIC

Now, let’s contrast this with a time when the conclusion of the Intermediate Value Theorem does not hold.

Consider the following situation,
PIC
and select all that are true:
is continuous on . is continuous on . is continuous on . is continuous on . There is a point in with .

Building on the question above, it is not difficult to see that each of the hypothesis of the Intermediate Value Theorem are 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 use to show that curves cross:

Now we move on to a more subtle example:

And finally, an example when the Intermediate Value Theorem does not apply.