[SOUND] Let's think back to our friend 
the square root function. 
What's the square root of two? 
Well, the square root of two is about 
1.414. 
What's the square root of 2.01? 
It's awfully close. 
It's 1.417 and a bit more, which is 
really close to the square root of two. 
What's the square root of 1.99? 
Well, it's also really close to the 
square root of two. 
It's 1.410 and a bit more, which is 
really close to 1.414. 
The point here is that nearby inputs are 
producing nearby outputs. 
Let's try to see these numbers. 
What does the graph of the square root 
function look like? 
Is this the graph of the square root 
function? 
Take a look at what happens right here. 
Nearby inputs are not being sent to 
nearby outputs. 
Inputs very close to two. 
Say something a little bit less than two 
and something a little bit bigger than 
two, are being sent to outputs that are 
quite far apart. 
The numbers show that couldn't have been 
the graph of the square root function. 
But what does the graph for the square 
root function look like? 
This is what the graph for the square 
root function looks like. 
It looks continuous, it's one nice curve. 
In particular, nearby inputs give rise to 
nearby outputs. 
Let's try to capture the concept of 
continuity. 
A bit more precisely than just a picture 
on the graph. 
Here's sort of a moral definition of what 
I mean when I say f of x is continuous at 
a. 
So morally I mean that inputs near a are 
being sent to outputs near f of a. 
From this perspective, it looks like the 
function, F of X equals the square root 
of X is continuous at two because inputs 
near two are being sent to outputs near 
the square root of two. 
How do we make this intuition a little 
bit more precise? 
Here is a precise definition, 
to say that F of X is continuous at A is 
to say that the limit of F of X as X 
approaches A is equal to F of A. 
Now think back to what we mean by limit, 
to say that the limit f of x equals f of 
a is to say that I can make f of x as 
close to f of a as you like as long as x 
is close enough to a. 
But that's really the spirit of 
continuity. 
Continuity is trying to say that nearby 
inputs are sent to nearby outputs, and 
this limit statement is capturing that 
sense. 
It's saying that I can make the output 
close to the output at A as long as the 
input is close to A. 
This definition's pretty involved. 
We've got to try to unpack this a bit. 
What does it really mean to say that the 
limit of F of X equals F of A as X 
approaches A? 
To make this statement I really need to 
know that f of x is defined at the point 
a. 
I can't talk about f of a unless I know 
that a is the domain of f. 
Talk about the limit of f of x I also 
need to know that the limit of f of x as 
x approaches a exists. 
I need this to be some number so I can 
talk about it being equal to some other 
number. 
Well once I've got these two statements, 
then it makes sense to claim that the 
limit of F of X as X approaches A is 
equal to F of A. 
But before I can make this third and 
final statement, I'm really assuming that 
these two preceding things hold, alright. 
So the definition of continuity is a 
little bit more subtle than it seems. 
It's really these three parts. 
The function has to be defined at the 
point. 
The limit has to exist and be equal to 
some number, and then I can say that 
number, the limit is equal to the 
functions value. 
So it makes sense to talk about the 
function at that point. 
Nothing we've done so far really captures 
the idea that the graph is a single 
curve, 
a single continuous curve. 
We've always just been working at a 
single point. 
This is the definition of continuity at 
the single point A. 
But often we want to talk about 
continuity on a whole interval at once. 
So we'll get rid of this and we'll make 
this solely a fancier definition. 
To say that the function is continuous on 
a whole interval from A to B is to say 
that for all points C in between A and B 
so C is bigger than A and less than B, 
so C is in the interval A to B. 
Then F of X is continuous at that point 
C. 
So this is what we mean when we talk 
about continuity on a whole interval at 
once. 
So that's the definition for open 
intervals. 
What about closed intervals? 
We have to be even a bit more careful 
when we talk about continuity on a closed 
interval, A B as opposed to the open 
interval A B. 
So if we say that the function is 
continuous on the closed interval from A 
to B. 
We mean that F of X is continuous on the 
open interval from A to B so in between A 
and B. 
But then what happens at A and at B? 
Well, the limit of F of X as X approaches 
A from the right hand side, is equal to F 
of A. 
And the limit of F of X as X approaches B 
from the left hand side is equal to F of 
B. 
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