1. Section 2.6 Differentiability

2. Consider the graph of f(x) = |x|
Is it continuous at x = 0?
Is it differentiable at x = 0?
Let’s zoom in at 0

6. No matter how close we zoom in, the graph never looks linear at x = 0
Therefore there is no tangent line there so it is not differentiable at x = 0
We can also demonstrate this using the difference quotient

7. The function f is differentiable at x if exists
Definition
The function f is differentiable at x if exists
Thus the graph of f has a non-vertical tangent line at x
We have 3 major cases
The function is not continuous at the point
The graph has a sharp corner at the point
The graph has a vertical tangent

8. Example

9. It has a vertical tangent at x = 0
Example
Note: This is a graph of
It has a vertical tangent at x = 0
Let’s see why it is not differentiable at 0 using our power rule

10. Example
Is the following function differentiable everywhere?
What values of a and b make the following function continuous and differentiable everywhere?