[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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