[SOUND] So far, we've been selling continuity as something about nearness.
Continuous functions preserve nearness, so nearby points gets into nearby points.
But continuity isn't just about how small changes become small changes.
Continuity has consequences for the global structure of the function as well.
Here's one of them. I've graphed just some random looking
continuous function. And I've picked a couple input points,
input point a, input point b.
Here is the point a, f of a, here's the point b, f of b.
And on the y-axis I've got the value f of a and the value f of b.
Now, in between f of a and f of b, I've just picked some random value.
I'm calling it y. Here's a consequence of continuity.
There has to be some corresponding input x so that if I plug in x, [SOUND] I get
out y. This is the so-called intermediate value
theorem. Let me write down a more precise
definition now. So here's a statement of the intermediate
value theorem. Theorem says the following.
Suppose f of x is continuous in the closed interval between a and b, and that
y is some point between f of a and f of b.
Then, there's an x between a and b so that the function's output, when you plug
in x, is equal to y. Here's a real world example, or if you
like, a real world non-example of the intermediate value theorem.
Is it possible for me to be standing here in one moment and over here in the next
moment without actually occupying the points in between?
What does the intermediate value theorem say?
My position is a continuous function of time.
So, if I'm standing here at say, time t equals zero seconds, and I'm over here at
say, time equals three seconds, mustn't there be a time when I'm
standing, say, right here? Yeah, maybe it's a time t equals two
seconds. Who knows?
What the intermediate value theorem says is that for a continuous function, all of
the intermediate values are actually achieved at some point along the way.
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