[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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