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STY

Studying Differentiability

Proposal: The absolute value function |x| is not differentiable at 0.

Before you go "wtf?", this is pretty confusing, but will make perfect sense at the end of this lecture.

So how do we even begin?

Well, the absolute value function, is "always positive".
We can split this into a piecewise function.

What is a piecewise function?

A function that's represented by two functions or more, for certain intervals. "Splitting" it in parts.

So, how do we split it?

Well, the absolute value function is "always non-negative"
So, for all negative x, we consider it as
For all x from zero and above, we just consider it as x. Cause, you know, it's not negative.

First: Check Continuity.

Do the negative number part first. Approach zero from the left.

That was easy. Now do that with the positive part. Approach zero from the right.

Well, look at what we have.

Let's prove if it's differentiable or not.

Let's do the negative number part first.

Do the same with the positive number part now. And put some of your favorite music that makes you feel like a badass, cause you're on a good path.

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Before getting into deeper topics, you'll learn a quick concept, then a little review:
The Continuity of The Derivative

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