functional equations - Multiplicative function on rationals

Let $\Bbb Q^+$ be the set of positive rational numbers.
Find all solutions $f:\Bbb Q^+ \to \Bbb R$ of the functional equation
$$
f(xy)=f(x)f(y), \quad x, y\in \Bbb Q.
$$

Is $f(x)=x^a$ the only solution?
If not, is it true if we assume that $f$ is continuous?