[MUSIC]. 
Remember, continuity is all how nearby 
inputs are sent to nearby outputs. 
Differentiability is how wiggling the 
input affects the output. 
In light of this, they seemed related, 
right? 
Something like the following seems 
plausible. 
Here's the theorem. 
Theorem. 
If f is differentiable at a, then f is 
continuous at a. 
In other words, a differentiable function 
is continuous. 
Morely, we know that a differentiable 
function is continuous. 
But were advanced enough at the is point 
in the course to give a precise argument 
using limit. 
Here we go. 
Let's suppose that f prime of a exists. 
In other words, that means a certain 
limit exists. 
What limit? Well, the limit of 
f(x)-f(x)/x-a as x approaches a. 
This limit of a difference quotients 
computes the derivative for the function 
at a. So, to say that the derivative 
exists is to say that this limit exists. 
Now, here comes the trick. 
What I'd like to compute is the limit of 
f(x)-f(a) as x approaches a, but I don't 
know how to do that directly. 
But, I can rewrite this thing I'm taking 
the limit of as a product. 
Watch. 
Instead of taking this limit, I'm going 
to take the limit as x approaches a of 
x-a times this difference quotient, times 
f(x)-f(a)/x-a. 
Now, as long as x isn't equal to a, this 
product is equal to this difference. 
Now, why does that help? Well, this is a 
limit of a product. 
So, by one of the limit laws, the limit 
of a product's the product of the limits 
as long as the limits exist. 
And in this case, they do. 
So, this limit of this product is, the 
product of the limits. 
It's the limit of x-a as x approaches a, 
times the limit of f(x)-f(a)/x-a. 
I'm only allowed to use this limit law 
because I know both of these limits 
exist. 
Now, this first limit, the limit of x-a 
as x approaches a, that's 0. 
And this second limit, well, this limit 
exists precisely because I'm assuming 
differentiability, 
the function's are differentiable. 
So, this limit is calculating the 
derivative at a, and zero times any 
number is equal to zero. 
The upshot here is that we've shown that 
the limit of f(x)-f(a)=0 as x approaches 
a. 
Why would you care about this? How does 
that help us? We know that the limit of 
f(x)-f(a) as x approaches a is equal to 
0. 
What that means is that the limit of f(x) 
as x approaches a is equal to f(a), but 
this is just the definition of 
continuity. 
So now we know that f is continuous at 
the point a. 
That's where we ended up. 
Remember, what we started with. 
We started by assuming that f was 
differentiable at a. 
And after doing all this work, we ended 
up concluding that f is continuous at the 
point a. 
So, differentiability implies continuity. 
One way to keep track of arguments like 
this is to think about clouds and rain. 
Theorem. 
If it is rainy, then it is cloudy. 
A shorter way of saying this, rainy 
implies cloudy. 
Now the question is, does it go the other 
way? If it's cloudy, is it necessarily 
rainy? Can you think of a cloudy day with 
no rain? Yes, today. 
Let's look out the window. 
It is very cloudy but there's no rain. 
Beyond clouds and rain, let's bring this 
back to the mathematics. A differentiable 
function is continuous, can you think of 
a continuous function which isn't 
differentiable? 
You might want to hit pause right now, 
if you don't want the puzzle given away. 
Here's an example of a function which is 
continuous but not differential, 
the function f(x)=|x|. 
We recently saw that the absolute value 
function wasn't differentable at zero. 
But how do we know that the absolute 
value function is continuous everywhere? 
We know that the absolute value function 
is continuous. 
I mean, look at it. 
It's all one piece. 
But, we can do better. 
We can use our limit knowledge to make a 
more precise argument. 
We know that f(x)=|x| is continuous for 
positive inputs. 
It's continuous on the open interval from 
zero to infinity because the function x 
is continuous there. 
And this function, the absolute value 
function, agrees with the function x if I 
plug in positive numbers. 
Likewise, I know that the function is 
continuous on negative inputs because the 
function -x is continuous there, and the 
function -x agrees with this function on 
this interval. 
The only sticking point is to check that 
the function's continuous at zero. And if 
I know it's continuous for positive 
inputs, negative inputs, and it's 
continuous at zero, then I know that it's 
continuous for all inputs. 
Now, how do I know that the absolute 
value function is continuous at zero? 
Well, that's another limit argument, 
right? The limit of the absolute value 
function when I push from the right-hand 
side is the same as the limit of the 
absolute value function when I push from 
the left-hand side, they're both zero. 
And because these two one-sided limits 
exist and agree, then I know the 
two-sided limit of the absolute value 
function is equal to zero, 
which is also the function's value at 
zero. 
And therefore, the abslute value function 
is continuous. 
In the end, there's some relationship 
between differentiability and continuity. 
Differentiable functions are continuous. 
Mathematics isn't just a sequence of 
unrelated concepts. 
[MUSIC] It's a single unified whole. 
All of these ideas are connected at the 
deepest possible levels. 
[MUSIC]