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[SOUND] Let's think back to our friend 
the square root function. 

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What's the square root of two? 
Well, the square root of two is about 

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1.414. 
What's the square root of 2.01? 

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It's awfully close. 
It's 1.417 and a bit more, which is 

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really close to the square root of two. 
What's the square root of 1.99? 

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Well, it's also really close to the 
square root of two. 

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It's 1.410 and a bit more, which is 
really close to 1.414. 

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The point here is that nearby inputs are 
producing nearby outputs. 

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Let's try to see these numbers. 
What does the graph of the square root 

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function look like? 
Is this the graph of the square root 

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function? 
Take a look at what happens right here. 

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Nearby inputs are not being sent to 
nearby outputs. 

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Inputs very close to two. 
Say something a little bit less than two 

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and something a little bit bigger than 
two, are being sent to outputs that are 

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quite far apart. 
The numbers show that couldn't have been 

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the graph of the square root function. 
But what does the graph for the square 

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root function look like? 
This is what the graph for the square 

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root function looks like. 
It looks continuous, it's one nice curve. 

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In particular, nearby inputs give rise to 
nearby outputs. 

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Let's try to capture the concept of 
continuity. 

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A bit more precisely than just a picture 
on the graph. 

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Here's sort of a moral definition of what 
I mean when I say f of x is continuous at 

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a. 
So morally I mean that inputs near a are 

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being sent to outputs near f of a. 
From this perspective, it looks like the 

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function, F of X equals the square root 
of X is continuous at two because inputs 

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near two are being sent to outputs near 
the square root of two. 

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How do we make this intuition a little 
bit more precise? 

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Here is a precise definition, 
to say that F of X is continuous at A is 

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to say that the limit of F of X as X 
approaches A is equal to F of A. 

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Now think back to what we mean by limit, 
to say that the limit f of x equals f of 

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a is to say that I can make f of x as 
close to f of a as you like as long as x 

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is close enough to a. 
But that's really the spirit of 

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continuity. 
Continuity is trying to say that nearby 

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inputs are sent to nearby outputs, and 
this limit statement is capturing that 

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sense. 
It's saying that I can make the output 

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close to the output at A as long as the 
input is close to A. 

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This definition's pretty involved. 
We've got to try to unpack this a bit. 

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What does it really mean to say that the 
limit of F of X equals F of A as X 

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approaches A? 
To make this statement I really need to 

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know that f of x is defined at the point 
a. 

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I can't talk about f of a unless I know 
that a is the domain of f. 

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Talk about the limit of f of x I also 
need to know that the limit of f of x as 

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x approaches a exists. 
I need this to be some number so I can 

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talk about it being equal to some other 
number. 

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Well once I've got these two statements, 
then it makes sense to claim that the 

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limit of F of X as X approaches A is 
equal to F of A. 

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But before I can make this third and 
final statement, I'm really assuming that 

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these two preceding things hold, alright. 
So the definition of continuity is a 

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little bit more subtle than it seems. 
It's really these three parts. 

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The function has to be defined at the 
point. 

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The limit has to exist and be equal to 
some number, and then I can say that 

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number, the limit is equal to the 
functions value. 

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So it makes sense to talk about the 
function at that point. 

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Nothing we've done so far really captures 
the idea that the graph is a single 

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curve, 
a single continuous curve. 

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We've always just been working at a 
single point. 

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This is the definition of continuity at 
the single point A. 

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But often we want to talk about 
continuity on a whole interval at once. 

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So we'll get rid of this and we'll make 
this solely a fancier definition. 

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To say that the function is continuous on 
a whole interval from A to B is to say 

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that for all points C in between A and B 
so C is bigger than A and less than B, 

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so C is in the interval A to B. 
Then F of X is continuous at that point 

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C. 
So this is what we mean when we talk 

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about continuity on a whole interval at 
once. 

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So that's the definition for open 
intervals. 

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What about closed intervals? 
We have to be even a bit more careful 

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when we talk about continuity on a closed 
interval, A B as opposed to the open 

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interval A B. 
So if we say that the function is 

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continuous on the closed interval from A 
to B. 

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We mean that F of X is continuous on the 
open interval from A to B so in between A 

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and B. 
But then what happens at A and at B? 

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Well, the limit of F of X as X approaches 
A from the right hand side, is equal to F 

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of A. 
And the limit of F of X as X approaches B 

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from the left hand side is equal to F of 
B. 

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[SOUND]