Name: Continuity over an interval | Limits and continuity | AP Calculus AB | Khan Academy
Uploaded: 2022-07-05T13:44:07.000Z
Duration: 9 min
Description: Thousands of YouTube videos with English-Chinese subtitles! Now you can learn to understand native speakers, expand your vocabulary, and improve your pronunciation...

To-infinitive

- [Instructor] What we're going to do in this video

The imperative

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

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

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

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

This is really just a more rigorous way of describing

this notion of not having to pick up your pencil,

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.

this shows that we're not including the endpoint.

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

but not equaling x equals a and x equals b.

So f is continuous over this open interval,

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

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

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

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,

and you actually had the definition of the function,

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

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

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

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

Let's think about the interval from negative two

So this is interesting because the function

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

with what exactly happens at negative two,

all the numbers larger than negative two.

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Continuity over an interval | Limits and continuity | AP Calculus AB | Khan Academy

approach

slightly

positive

immediately