Assume is a function defined for all such that
For parts (b)-(d) assume also that is differentiable for all real (i.e., assume exists everywhere).
- Using the given equation above, prove or . Prove that if then for all .
- Prove that for all .
- Prove that for some constant .
- Prove that if then .
- Proof. Assume . Then if by the functional equation we have
Therefore, if then we must have . Hence, we have either or .
Furthermore, if , then for any by the functional equation we have
Hence, for any
- Proof. Since the derivative of exists everywhere by hypothesis, we use the limit definition of the derivative and the functional equation to write,
- Proof. First, if then by the functional equation we know for any we have
So, if then is the constant function which is 0 everywhere. Hence, for all .
Next, we consider the case that . Since exists for all we use the definition of derivative and the functional equation again to write,
The final line follows since exists by assumption so we can write for some
- Proof. Since (by part (c)) we know, by the previous exercise (Section 6.17, Exercise #39) that for some constant . Since by assumption, we know by part (a) that . Hence,