Discontinuities of The Second Kind - The Cusp

Now let's consider as an example the function

Let's take its derivative:

Before you look at the solution, try to compute the limit to zero of this derivative.

Let's start with the limit from the left.

A cusp is a discontinuity in which the derivative at a point tends to two different infinities.

This is what causes the distinct "pinched" look in the graph of a function with a cusp at a point.

Before going in deeper into Function Analysis, we need to learn some more concepts and rules.
Introducing your newest nightmare:
The Chain Rule

Interactive Graph