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,