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[SOUND] So far, we've been selling 
continuity as something about nearness. 

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Continuous functions preserve nearness, 
so nearby points gets into nearby points. 

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But continuity isn't just about how small 
changes become small changes. 

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Continuity has consequences for the 
global structure of the function as well. 

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Here's one of them. 
I've graphed just some random looking 

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continuous function. 
And I've picked a couple input points, 

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input point a, 
input point b. 

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Here is the point a, f of a, 
here's the point b, f of b. 

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And on the y-axis I've got the value f of 
a and the value f of b. 

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Now, in between f of a and f of b, I've 
just picked some random value. 

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I'm calling it y. 
Here's a consequence of continuity. 

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There has to be some corresponding input 
x so that if I plug in x, [SOUND] I get 

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out y. 
This is the so-called intermediate value 

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theorem. 
Let me write down a more precise 

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definition now. 
So here's a statement of the intermediate 

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value theorem. 
Theorem says the following. 

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Suppose f of x is continuous in the 
closed interval between a and b, and that 

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y is some point between f of a and f of 
b. 

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Then, there's an x between a and b so 
that the function's output, when you plug 

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in x, is equal to y. 
Here's a real world example, or if you 

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like, a real world non-example of the 
intermediate value theorem. 

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Is it possible for me to be standing here 
in one moment and over here in the next 

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moment without actually occupying the 
points in between? 

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What does the intermediate value theorem 
say? 

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My position is a continuous function of 
time. 

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So, if I'm standing here at say, time t 
equals zero seconds, and I'm over here at 

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say, time equals three seconds, 
mustn't there be a time when I'm 

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standing, say, right here? 
Yeah, maybe it's a time t equals two 

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seconds. 
Who knows? 

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What the intermediate value theorem says 
is that for a continuous function, all of 

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the intermediate values are actually 
achieved at some point along the way. 

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[SOUND]