What is differentiability?

Differentiability is when a function has a defined derivative. If a function's derivative is undefined at a certain point, it is called non-differentiable, or indifferentiable at that point.

Types of indifferenciable points

There are 3 types of indifferenciable points:

1. cusp

2. vertical tangent

3. discontinuity

1. indifferentiable by cusp

You can think of a cusp as a sharp point, or a "corner" on a graph. Such points do not have a tangent that can be drawn on them, and thus are not differenciable.

Here are two examples of graphs with an indifferenciable cusp:

2. indifferentiable by vertical tangent

A vertical tangent does not exist. This is because the slope of a vertical tangent is $\infty$, which is not a real number.

Here is an example:

3. indifferentiable by discontinuity

This is fairly obvious, if the function is discontinuous, it cannot be differentiable.

By looking at the previous discontinuous examples, you can see that that the function $f(x)$ is not differentiable when $x = 2$ in the first graph, and when $x \leq 2$ in the second graph.