abstract algebra - Functions over $R$ such that $f(xy) = f(x)f(y)$

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)?