is the intermediate value theorem.

Which, despite some of this mathy language you'll see

is one of the more intuitive theorems

possibly the most intuitive theorem you will

come across in a lot of your

mathematical career.

So first I'll just read it out

and then I'll interpret it and hopefully

we'll all appreciate that it's pretty obvious.

I'm not going to prove it here.

But, I think the conceptual underpinning here is

it should be straightforward.

So the theorem tells us that suppose F is a function

continuous at every point of the interval

the closed interval, so we're including A and B.

So it's continuous at every point of the interval A, B.

Let me just draw a couple of examples

of what F could look like just

based on these first lines.

Suppose F is a function continuous at every point

of the interval A, B.

So let me draw some axes here.

So that's my Y axis.

And this is my X axis.

So, one situation

if this is A.

And this is B.

F is continuous at every point of the interval

of the closed interval A and B.

So that means it's got to be for sure defined

at every point.

As well, as to be continuous you have

to defined at every point.

And the limit of the function that is recorded at that point

should be equal to the value of the function of that point.

And so the function is definitely going to be defined

at F of A.

So it's definitely going to have an F of A

right over here.

That's right over here

is F of A.

Maybe F of B is higher.

Although we can look at different cases.

So that would be our F of B.

And they tell us it is a continuous function.

It is a continuous function.

So if you're trying to imagine continuous functions

one way to think about it is

if we're continuous over an interval

we take the value of the function at

one point of the interval.

And, if it's continuous we need to be able to

get to the other, the value of the function

at the other point of the interval

without picking up our pencil.

So, I can do all sorts of things

and it still has to be a function.

So, I can't do something like that.

But,

as long as I don't pick up my pencil

this is a continuous function.

So, there you go.

If the somehow the graph

I had to pick up my pencil.

If I had to do something like this

oops, I got to pick up my pencil do something like that,

well that's not continuous anymore.

If I had to do something like this

and oops, pick up my pencil

not continuous anymore.

If I had to do something like

wooo.

Whoa, okay, pick up my pencil, go down here,

not continuous anymore.

So, this is what a continuous function

that a function that is continuous

over the closed interval A, B looks like.

I can draw some other examples, in fact,

let me do that.

So let me draw

one.

Maybe where F of B is less than

F of A.

So it's my Y axis.

And this is my

X axis.

And once again, A and B don't both

have to be positive,

they can both be negative.

One could be, A could be negative.

B could be positive.

And maybe in this situation.

And F of A and F of B

it could also be a positive or negative.

But let's take a situation where this is

F of A.

So that, right over there,

is F of A.

This right over here

is F of B.

F of B.

And once again we're saying F is a

continuous function.

So I should be able to go from F of A

to F of B

F of B draw a function

without having to pick up my pencil.

So it could do something like this.

Actually I want to make it go vertical.

It could go like this

and then go down.

And then

do something

like that.

So these are both cases

and I could draw an infinite number of cases

where F is a function continuous at every point

of the interval.

The closed interval, from A to B.

Now, given that

there's two ways to state the conclusion

for the intermediate value theorem.

You'll see it written in one of these ways

or something close to one of these ways.

And that's why I included both of these.

So one way to say it is, well

if this first statement is true

then F will take on every value

between F of A and F of B

over the interval.

And you see in both of these cases

every interval, sorry, every

every value between F of A

and F of B.

So every value here

is being taken on at some point.

You can pick some value.

You can pick some value, an arbitrary value

L, right over here.

Oh look.

L happened right over there.

If you pick L

well, L happened right over there.

And actually it also happened there

and it also happened there.

And this second bullet point describes

the intermediate value theorem more that way.

For any L between the values of F and A

and F of B

there are exists a number C

in the closed interval from A to B

for which F of C equals L.

So there exists at least one C.

So in this case

that would be our C.

Over here, there's potential

there's multiple candidates for C.

That could be a candidate for C.

That could be a C.

So we could say there exists at least

one number.

At least

one number, I'll throw that in there,

at least one number C

in the interval for which this is true.

And, something that might amuse you

for a few minutes is

try to draw a function where this first

statement is true.

But somehow the second statement is

not true.

So, you say, okay, well let's say

let's assume that there's an L

where there isn't a C in the interval.

Let me try and do that.

And I'll draw it big so that

we can really see how obvious

that we have to take on

all of the values between F and A

and F of B is.

So,

let me draw a big axis this time.

So that's my Y axis.

And,

that is my X axis.

And I'll just do the case where

just for simplicity, that is A

and that is B.

And let's say

that this is F of A.

So that is

F of A.

And let's say that this

is F of B.

Little dotted line.

All right.

F of B.

And we assume that we

we have a continuous function here.

So the graph, I could draw it from

F of A to F of B from this point

to this point

without picking up my pencil.

From this coordinate A comma F of A

to this coordinate B comma F of B

without picking up my pencil.

Well, let's assume

that there is some L that we don't take on.

Let's say there's some value L right over here.

And,

and we never take on this value.

This continuous function never takes on this value

as we go from X equaling A to X equal B.

Let's see if I can draw that.

Let's see if I can get

from here

to here

without ever essentially crossing this dotted line.

Well let's see, I could, wooo,

maybe I would a little bit.

But gee, how am I gonna get there?

Well, without picking up my pencil.

Well,

well, I really need to cross that line,all right.

Well, there you go.

I found, we took on the value L

and it happened at C

which is in that closed interval.

So once again, I'm not giving you a proof here.

But hopefully you have a good intuition

that the intermediate value theorem

is kind of common sense.

The key is you're dealing with a continuous function.

If you make its graph

if you were to draw it between

the coordinates A comma F of A

and B comma F of B

and you don't pick up your pencil,

which would be true of a continuous function.

Well, it's going to take on every value

between F of A and F of B.