0:00

[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|.

4:03

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]