This question already has an answer here:
I had a discussion with a friend and there it came up the question whether $f(x)f(y)=f(x+y)$, $f(0)=1$ and the existence of $f'(x)$ implies that $f(x)=\exp(a x)$. This seems very reasonable but I cannot figure this out. Can differentiability be pushed to smoothness in this case?
In a similar manner on would expect that $g(x)+g(y)=g(xy)$ and $g(1)=0$ would imply that $g(x)=\log(a x)$.