Possible Duplicate:
If $f(xy)=f(x)f(y)$ then show that $f(x) = x^t$ for some t
Solution for exponential function's functional equation by using a definition of derivative
I can think of three functions that satisfy the condition $f(xy) = f(x)f(y)$ for all $x, y$, namely
- $f(x) = x$
- $f(x) = 0$
- $f(x) = 1$
Are there more?
And is there a good way to prove that such a list is exhaustive (once expanded to include any other examples that I haven't thought of)?