Given a function satisfying the properties:
and
Prove the following:
- The derivative
exists for all
.
- We must have
.
This problem is quite similar to two previous exercises here and here (Section 6.17, Exercises #39 and #40).
- Proof. To show that the derivative
exists for all
we must show that the limit
exists for all
. Using the given properties of
we can evaluate this limit
Therefore,
for all
, so the derivative is defined everywhere
- Proof. From part (a) we know
. By Section 6.17, Exercise #39 (linked above) we know that the only functions
which satisfy this equation are
for all
or
for some constant
(where
in the linked exercise). However, since the derivative of
exists everywhere, and differentiability implies continuity, we know
is continuous everywhere. Hence,
. Then,
since
, so
. Therefore, we must have
for some constant
. Furthermore, we must have
since
. Thus,