Let be a function which is differentiable everywhere and which satisfies

for some positive constants and . What can you conclude about such a function ?

(**Note:** I’m not entirely sure what Apostol wants here since the instruction “what can you conclude” is pretty vague. He does give an “answer” in the back of the book, so I verify that it does have the properties indicated, but I don’t know how you would arrive at that expression just from the question statement. I’ll mark this question as incompletely and hopefully come up with something better in the future.)

Since satisfies the functional equation we can write

which implies

Then computing

Thus, is indeed periodic with period and so

So, this definition of in terms of the periodic function indeed satisfies the functional equation.

I think you can derive the form from the back of the book. Assuming f satisfies the functional equation, we may define

g(x)=b^(-x/a)f(x), and we then find that

g(x+a)=b^((-x/a)-1)f(x+a)=b^((-x/a)-1)*bf(x)=b^(-x/a)f(x),

and conclude that g is periodic with period a.

This is circular reasoning, you used the fact that f(x+a)=bf(x) to prove that g is periodic and then used that to prove that f(x+a)=bf(x) !!!