[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]