is explore continuity over an interval.

But to do that, let's refresh our memory

about continuity at a point.

So we say that f

is continuous

when

x is equal to c,

if and only if,

so I'm gonna make these two-way arrows right over here,

the limit

of f of x

as x approaches c

is equal to f of c.

And when we first introduced this,

we said, hey, this looks a little bit technical,

but it's actually pretty intuitive.

Think about what's happening.

The limit as x approaches c of f of x,

so let's say that f of x as x approaches c

is approaching some value.

So if we approach,

if we approach from the left,

we're getting to this value.

If we approach from the right, we're getting this value.

Well, in order for the function to be continuous,

if I had to draw this function without picking up my pen,

well, the value of the function at that point

should be the same as the limit.

This is really just a more rigorous way of describing

this notion of not having to pick up your pencil,

this notion of connectedness,

that you don't have any jumps

or any discontinuities of any kind.

So with that out the way,

let's discuss continuity over intervals.

Let me delete this really fast,

so I have space to work with.

So we say, so I'm gonna first talk about an open interval,

and then we're gonna talk about a closed interval

because a closed interval gets a little bit more involved.

So we say f

is continuous

over an open interval from a to b.

So the parentheses instead of brackets,

this shows that we're not including the endpoint.

So this would be all of the points between x equals a

and x equals b,

but not equaling x equals a and x equals b.

So f is continuous over this open interval,

if and only if,

if and only if,

f is continuous,

f is continuous

over

every point in,

over every point

in

the interval.

So let's do a couple of examples of that.

So let's say we're talking about the open interval

from negative seven

to negative five.

Is f continuous over that interval?

Let's see, we're going from negative seven

to negative five,

and there's a couple of ways you could do it.

There's the not-so-mathematically-rigorous way,

where you could say, hey, look, if I start here,

I can get all the way to negative five

without having to pick up my pencil.

If you wanted to do more rigorously

and you actually had the definition of the function,

you might be able to do a proof,

that for any of these points over the interval,

that the limit as x approaches any one of these points

of f of x is equal to the value of the function

at that point.

It's harder to do when you only have a graph.

When you only have a graph,

you can only just do it by inspection,

and say, okay, I can go from that point to that point

without picking up my pencil,

so I feel pretty good about it.

Now let's do another interval.

Let's say the, so let me put a check mark here,

that is continuous.

Let's think about the interval from negative two

to positive one, the open interval.

So this is interesting because the function

at negative two is up here.

And so if you really wanted to start at negative two,

you would have to start here and then jump immediately down

as soon as you get slightly larger than negative two

and then keep going.

But this is an open interval,

so we're not actually concerned

with what exactly happens at negative two,

we're concerned what happens when we are

all the numbers larger than negative two.

So we would actually start right over here,

and then we would go to one.

And once again, based on the intuitive

I didn't have to pick up my pen idea,

this function would be continuous

over this,

over this interval.

So what's an example of an interval

where the function would not be continuous?

Well, think about the interval from,

well, this is a pretty straightforward one,

the open interval from three to five.

The function is here when x is equal to three.

But if we wanted to get to five,

it looks like we're asymptoting,

it looks like we're asymptoting up towards infinity

and we just keep on going for a very long time.

And then we would have to pick up our pencil and jump over,

and then we would come back down right over here.

And so here we are not continuous over that interval.

So now let's think about the more,

the slightly more involved interval.

The slightly more involved case

is when you have a closed interval.

F is continuous

over the closed interval from a to b.

So this includes not just the points between a and b,

but the endpoints as well,

if and only if,

f is

continuous

over the open interval

and the one-sided limits.

Let me right this.

And

the limit

as x approaches a

from the right

of f of x

is equal to f of a,

and the limit

as x approaches b

from the left,

from the left of f of x

is equal to f of b.

Now what's going on here?

Well, it's just saying that the one-sided limit,

when you're operating within the Interval,

has to approach the same value as the function.

So for example, if we said the closed interval

from negative seven to negative five,

well, this one is still reasonable,

you know, just based on the picking up your pencil thing.

You don't have to pick up your pencil.

And what you would do is at the endpoint,

and at negative seven,

this function is just plain old continuous,

but if it wasn't defined over here,

it could still be continuous because you would do

the right-handed limit towards it.

And you'd say, okay, the right-handed limit

is equal to the value of the function.

And then at this endpoint, at the second endpoint,

you'd say, okay, the left-handed limit

is equal to the function, even if it wasn't defined here,

even if the two-sided limit were not defined.

And so we could actually look at an example of that.

If we were looking at

the interval from the closed,

and you could have one side open, one side closed,

but let's just do the closed interval from negative three

to negative two.

So notice I did not have to pick up my pencil.

I'm including negative three,

and I'm getting all the way to negative two.

If you knew the analytic definition of this function,

you could prove that, hey, the limit at any of these points

inside, between negative three and negative two,

is equal to the value of the function.

Negative three, the function is clearly, at negative three,

the function is just plain old continuous.

The two-sided limit approaches the value of the function.

But at negative two, the two-sided limit does not exist.

When you approach from the left,

it looks like you're approaching zero.

F of x is equal to zero.

When you approach from the right,

it looks like f of x is approaching negative three.

So even though the two-sided limit does not exist,

we can still be good

because the left-handed limit does exist.

And the left-handed limit is approaching

the value of the function.

So we actually are continuous over that interval.

But then if we did the interval,

if we did the closed interval from negative two to

negative two to one,

pause the video and think about,

based on what we just talked about,

are we continuous over this interval?

Well, we're going from negative two

to one,

and negative two is the lower bound.

So is this right over here,

is this right over here true?

Is the limit as we approach negative two from the right,

is that the same thing as f of negative two?

Well, the limit as we approach from the right

seems to be approaching negative three,

and f of negative two is zero.

So this limit does not, this, this,

these two things, the limit as we approach from the right

and the value of the function are not the same.

And so we do not have that, I guess you could say

that one-sided (laughs) continuity at negative two.

And that also makes sense.

If I start at negative two,

let me do this in a color you can see,

if I start at negative two

and I want to go the rest of the interval to one,

I have to pick up my pencil.

Pick up my pencil, go here, and then keep on going.

So this is, we are not continuous over that interval.