
4-6 Calculus – Differentiation and Integration
CONVERSE
The converse of the theorem is false. That is, a continuous function need not be differentiable.
Example: Consider the function f(x) = |x|. We know that f is continuous at 0.
But f is not differentiable at 0 because
f '
x h x
h
x h x
h
h
h
( ) lim
lim
0
1
0
0
+
→
→
=
+ −
=
+ −
=
+
+
and
f '
x h x
h
x h x
h
f ' f '
h
h
( ) lim
lim
( )
.
( ) ( ).
0
1
0 0
0
0
−
→
→
+ −
=
+ −
=
− + −
= −
∴ ≠
−
+
4.1.1 Derivative as the Rate of Change
The concept of derivative can also be described as a method to compute the rate at which a
dependent output y changes with respect to the change in the independent input x. This rat