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[MUSIC]. 
Remember, continuity is all how nearby 

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inputs are sent to nearby outputs. 
Differentiability is how wiggling the 

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input affects the output. 
In light of this, they seemed related, 

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right? 
Something like the following seems 

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plausible. 
Here's the theorem. 

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Theorem. 
If f is differentiable at a, then f is 

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continuous at a. 
In other words, a differentiable function 

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is continuous. 
Morely, we know that a differentiable 

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function is continuous. 
But were advanced enough at the is point 

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in the course to give a precise argument 
using limit. 

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Here we go. 
Let's suppose that f prime of a exists. 

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In other words, that means a certain 
limit exists. 

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What limit? Well, the limit of 
f(x)-f(x)/x-a as x approaches a. 

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This limit of a difference quotients 
computes the derivative for the function 

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at a. So, to say that the derivative 
exists is to say that this limit exists. 

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Now, here comes the trick. 
What I'd like to compute is the limit of 

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f(x)-f(a) as x approaches a, but I don't 
know how to do that directly. 

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But, I can rewrite this thing I'm taking 
the limit of as a product. 

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Watch. 
Instead of taking this limit, I'm going 

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to take the limit as x approaches a of 
x-a times this difference quotient, times 

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f(x)-f(a)/x-a. 
Now, as long as x isn't equal to a, this 

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product is equal to this difference. 
Now, why does that help? Well, this is a 

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limit of a product. 
So, by one of the limit laws, the limit 

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of a product's the product of the limits 
as long as the limits exist. 

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And in this case, they do. 
So, this limit of this product is, the 

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product of the limits. 
It's the limit of x-a as x approaches a, 

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times the limit of f(x)-f(a)/x-a. 
I'm only allowed to use this limit law 

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because I know both of these limits 
exist. 

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Now, this first limit, the limit of x-a 
as x approaches a, that's 0. 

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And this second limit, well, this limit 
exists precisely because I'm assuming 

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differentiability, 
the function's are differentiable. 

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So, this limit is calculating the 
derivative at a, and zero times any 

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number is equal to zero. 
The upshot here is that we've shown that 

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the limit of f(x)-f(a)=0 as x approaches 
a. 

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Why would you care about this? How does 
that help us? We know that the limit of 

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f(x)-f(a) as x approaches a is equal to 
0. 

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What that means is that the limit of f(x) 
as x approaches a is equal to f(a), but 

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this is just the definition of 
continuity. 

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So now we know that f is continuous at 
the point a. 

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That's where we ended up. 
Remember, what we started with. 

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We started by assuming that f was 
differentiable at a. 

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And after doing all this work, we ended 
up concluding that f is continuous at the 

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point a. 
So, differentiability implies continuity. 

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One way to keep track of arguments like 
this is to think about clouds and rain. 

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Theorem. 
If it is rainy, then it is cloudy. 

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A shorter way of saying this, rainy 
implies cloudy. 

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Now the question is, does it go the other 
way? If it's cloudy, is it necessarily 

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rainy? Can you think of a cloudy day with 
no rain? Yes, today. 

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Let's look out the window. 
It is very cloudy but there's no rain. 

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Beyond clouds and rain, let's bring this 
back to the mathematics. A differentiable 

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function is continuous, can you think of 
a continuous function which isn't 

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differentiable? 
You might want to hit pause right now, 

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if you don't want the puzzle given away. 
Here's an example of a function which is 

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continuous but not differential, 
the function f(x)=|x|. 

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We recently saw that the absolute value 
function wasn't differentable at zero. 

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But how do we know that the absolute 
value function is continuous everywhere? 

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We know that the absolute value function 
is continuous. 

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I mean, look at it. 
It's all one piece. 

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But, we can do better. 
We can use our limit knowledge to make a 

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more precise argument. 
We know that f(x)=|x| is continuous for 

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positive inputs. 
It's continuous on the open interval from 

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zero to infinity because the function x 
is continuous there. 

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And this function, the absolute value 
function, agrees with the function x if I 

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plug in positive numbers. 
Likewise, I know that the function is 

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continuous on negative inputs because the 
function -x is continuous there, and the 

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function -x agrees with this function on 
this interval. 

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The only sticking point is to check that 
the function's continuous at zero. And if 

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I know it's continuous for positive 
inputs, negative inputs, and it's 

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continuous at zero, then I know that it's 
continuous for all inputs. 

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Now, how do I know that the absolute 
value function is continuous at zero? 

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Well, that's another limit argument, 
right? The limit of the absolute value 

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function when I push from the right-hand 
side is the same as the limit of the 

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absolute value function when I push from 
the left-hand side, they're both zero. 

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And because these two one-sided limits 
exist and agree, then I know the 

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two-sided limit of the absolute value 
function is equal to zero, 

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which is also the function's value at 
zero. 

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And therefore, the abslute value function 
is continuous. 

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In the end, there's some relationship 
between differentiability and continuity. 

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Differentiable functions are continuous. 
Mathematics isn't just a sequence of 

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unrelated concepts. 
[MUSIC] It's a single unified whole. 

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All of these ideas are connected at the 
deepest possible levels. 

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[MUSIC]