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,