PROBLEM:

Suppose that a function satisfies the following conditions for all real values and :

i.

ii. , where

Show that the derivative exists at every value of and that .

I honestly don't have much of a clue here. Instead, I've tried to ask myself questions to get some idea going but to no avail. First, I tried to make sense of what the conditions imply and discovered that because and belong in the same set of real numbers without restriction, then everything said about is also true about .

ATTEMPT:

Therefore, , where

I thought, maybe, if I could rewrite condition (i) with a substitution in terms of (ii), I might get somewhere.

Any thoughts?