I am going back through a bunch of calculus I learned in high school and proving the stuff that they just told us was true. Along the way, I found I had to prove that if $f(x+y)=f(x)f(y)$ then $f$ is an exponential function.
I managed to do it by induction on the integers, then generalizing to the rationals by the fundamental theorem of arithmetic, then to the reals by continuity. It involved some rather ugly analysis for $f(1/2)$ because there are two possible values for the square root, which I didn't have to worry about for the other primes because, over $R$, odd powers are invertible. But this prevents my argument from being generalized to the complex numbers.
There has to be a more elegant way to get this result. I'm hoping you can point me in the right direction toward a better proof (preferably without giving the whole thing away -- the point, after all, is to develop my ingenuity).