Section 4.2 Differentiability
In the previous section we reviewed that the derivative of a function is defined as a limit. But when we studied limits, we realized that sometimes limits do not exist. What happens if the limit in the definition of the derivative of a given function does not exist at a point? This is what we study in this section. Basically, a function \(f(x)\) is “differentiable” at a point \(x=a\) if the limit in the definition of its derivative \(f'(a)\) exists. Let us now study this in more detail!
Objectives
You should be able to:
- Explain and illustrate the definition of differentiability.
- Apply the definition of differentiability to determine whether or not a function is differentiable at a point \(a\text{.}\)
- Visually identify where functions are not differentiable (corners, discontinuities, vertical tangents), and explain why.
- Relate the notions of differentiability and continuity.
Subsection 4.2.1 Instructional video
There is no instructional video on this particular topic. It will be covered during the live lecture.
Subsection 4.2.2 Key concepts
Concept 4.2.1. Differentiability.
A function \(f\) is differentiable at \(x=a\) if \(f'(a)\) exists (as a limit). It is differentiable on an open interval \((a,b)\) if it is differentiable at every point in the interval.
Concept 4.2.2. Relation between continuity and differentiability.
A function \(f\) that is not continuous at \(x=a\) is also not differentiable at \(x=a\text{.}\) Equivalently, a function \(f\) that is differentiable at \(x=a\) must be continuous at \(x=a\text{.}\)
The converse is not true: a function \(f\) that is continuous at \(x=a\) may not be differentiable at \(x=a\) (think of \(f(x) = |x|\) at \(x=0\)).
Concept 4.2.3. Where functions fail to be differentiable.
A function can fail to be differentiable at \(x=a\) in three different ways:
- It is not continuous at \(x=a\text{;}\)
- The left-sided limits and right-sided limits of the difference quotient are not the same, in which case \(f\) has a corner or a kink at \(x=a\text{;}\)
- The limit of the difference quotient is infinite, in which case \(f\) has a vertical tangent at \(x=a\text{.}\)