[SOUND] Here's an application of the 
intermediate value theorem I'm very fond 
of. 
Here's how it goes. 
Suppose you've got a function f. 
f is continuous in the closed interval 
between zero and one. 
And whenever x is between zero and one, f 
of x is between zero and one. 
So that's what you've got to suppose when 
you get out of this. 
Then, there is a point x, x between zero 
and one, and f of x = x. 
It's telling you that there exists an 
input. 
So that if you apply f to that input 
value, you get the output which is the 
same as the thing you plugged in. 
You call these things fixed points. 
You imagine that f is moving around the 
points between zero and one, 
and you're finding a point x which 
doesn't move a fixed point. 
Why is something like this true? 
Let's see. 
Let me use the Intermediate Value Theorem 
but I'm not going to apply the 
Intermediate Value Theorem to f but I'm 
going to make up a new function. g(x) = 
f(x) - x. 
What do I know about g? 
g's continuous. 
f is continuous by assumption. 
The identity function x is continuous and 
differences of continuous functions are 
continuous. 
So, g is a continuous function and the 
closed interval 01. 
I also know some of the values of g. 
What's g of zero? 
Well, g(0) is by definition f(0) - 0. 
f(0) is between zero and one. 
So I've got a number between zero and one 
minus zero, 
that number's at least zero. 
I also know something about g(1). 
g(1) is by definition f(1) - 1 f(1) is 
between zero and one, it could be zero, 
could be one. 
But, a number between zero and one minus 
one is less than or equal to zero. 
What do I know altogether? 
I've got a continuous function g, its 
value at zero is bigger than equal to 
zero. 
Its value at one is less than equal to 
zero. 
By the intermediate value theorem, this 
gives me a point x so that g(x) = 0. 
Why do I care about finding a point x so 
that g(x) is equal to zero? 
Well, if I take a look at what g(x) is 
equal to, g(x) is f(x) - x. 
So, I found a point x so that f(x) - x is 
equal to zero. 
I found a point x so that f(x) = x, I 
found the fixed point. 
So, we've seen why this is true. 
Let'd try to apply this to a specific 
example life f(x) equals cosine x. 
Now, here I've got a graph of cosine and 
that the graph of y equals cosine x. 
Now, what do I know about cosine? 
I know cosine's continuous and cosine of 
a number between zero and one is a number 
between zero and one. 
So, our statement applies to it. 
There's a fixed point for cosine. 
There's some value of x between zero and 
one so cosine of x equals x. 
Let's see if we can find that on a 
computer. 
Okay. 
I've got cosine loaded up on to my 
computer. 
Cosine of one is about 0.54. 
Let me decrease the input. 
And I notice that cosine of 0.68 is 0.77. 
Here the input is smaller than the 
output. 
Let me increase the input. 
Now, notice what happened. 
Now the input is 0.74 but the output is 
smaller than 0.74. 
Cosine of 0.74 is just 0.73. 
Now, it decreased the input. 
Oh, the opposite thing happened. 
Now the input is smaller than the output. 
Now, I'll increase the input. 
I'll decrease the input. 
Increase the input. 
And you can see that I, I can sort of 
zoom in or narrow in on the correct value 
of of x so that cosine of x is equal to 
x. 
Sees in the computer I found that this 
point here is at about 0.739. 
So, without a number between zero and 
one, cosine of that number is itself. 
It's a fixed point for cosine. 
These kinds of fixed point theorem 
statements that assert the existence of 
fixed points, 
they pervade mathematics. 
Here's another example. 
I've got a map of Ohio right here. 
Let me apply a continuous function to 
this map. 
[SOUND] I'm going to throw it down to the 
ground, 
stomp on it. 
And now let's take a look at it. 
[SOUND] The question is, is there a point 
on the map which is exactly above its 
corresponding point in the ground? 
And indeed there is. 
I'll use this pin to stab through that 
point. 
[SOUND]