Find all real numbers a for which there exists a non-constant function \( f: \mathbb{R} \rightarrow \mathbb{R} \) satisfying the following two equations for all \( x \in \mathbb{R} \):
i) \( \quad f(a x)=a^{2} f(x) \) and

ii) \( f(f(x))=a f(x) \).
The conditions of the problem give two representations for $f(f(f(x)))$:

$$
f(f(f(x)))=a f(f(x))=a^{2} f(x)
$$

and

$$
f(f(f(x)))=f(a f(x))=a^{2} f(f(x))=a^{3} f(x).
$$

So $a^{2} f(x)=a^{3} f(x)$ for all $x$, and if there is an $x$ such that $f(x) \neq 0$, then $a=0$ or $a=1$. Otherwise $f$ is the constant function $f(x)=0$ for all $x$. If $a=1$, the function $f(x)=x$ satisfies the conditions. For $a=0$, one possible solution is the function $f$,

$$
f(x)=\left\{\begin{array}{ll}
1 & \text { for } x<0 \\
0 & \text { for } x \geq 0
\end{array} .\right.
$$