Sec. 3.2: Working with the DerivativeDifferentiability and Continuity

This is used only in special cases (i.e. functions that are continuous but not differentiable).

Sec. 3.2: Working with the DerivativeDifferentiability and Continuity

The existence of the limit in this alternate form requires that the one-sided limits

exist and are equal.

Sec. 3.1: Introducing the Derivative

Differentiability and ContinuityEx: Discuss the differentiability of

f(x) = |x – 2| at x = 2.

limx c

f x f cf x

x c

2

2lim

2x

f x fx

2

2 2 2lim

2x

xx

2

2lim

2x

xx

2

2lim 1

2x

xx

2

2lim 1

2x

xx

2 2

2 2lim lim

2 2x x

x xx x

f is not differentiable at x = 2.

Sec. 3.2: Working with the Derivative

Differentiability and ContinuityEx: Discuss the differentiability of

f(x) = x1/3 at x = 0.

limx c

f x f cf x

x c

0

0lim

0x

f x fx

1 3

0

0limx

xx

2 30

1limx x

2 30

1lim0x

2 30

1lim0x

f has a vertical tangent at x = 0.

Sec. 3.1: Introducing the DerivativeDifferentiability and Continuity

Sec. 3.2: Working with the DerivativeDifferentiability and ContinuitySummary of the relationship between differentiability

and continuity:

1. Differentiability implies continuity. (A function that is differentiable must also be continuous.)

2. Continuity does not imply differentiability. (A function that is continuous is not necessarily differentiable.)• Things that destroy differentiability.a) Discontinuityb) Sharp turns (cusps)c) Vertical tangents