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
