Differentiability

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!

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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.

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Subsection 4.2.1 Instructional video

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There is no instructional video on this particular topic. It will be covered during the live lecture.

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Subsection 4.2.2 Key concepts

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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.

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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 $x=0$ ).

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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 $x=a\text{.}$

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MATH 144: Calculus for the Physical Sciences I