Understand the problem

Find all real numbers $a$ for which there exists a non-constant function $f :\Bbb R \to \Bbb R$ satisfying the following two equations for all $x\in \Bbb R:$
i) $f(ax) = a^2f(x)$ and
ii) $f(f(x)) = a f(x).$
Source of the problem
Baltic Way 2016
Topic
Functional Equations
Difficulty Level
Easy
Suggested Book
Functional Equations by BJ Venkatachala

Start with hints

Do you really need a hint? Try it first!

Show that the choices a=0,1 work.

Show that af(f(x))=a^2f(f(x)). As we have already dealt with a=0, this gives af(f(x))=f(f(x)).

Hint 3 gives (a-1)f(f(x))=0. As a=1 has already been dealt with, we must consider the option f(f(x))\equiv 0.
Hint 3 gives af(x)\equiv 0. As a\neq 0, we have f(x)\equiv 0. This contradicts the fact that f is non-constant. Hence, a=0,1 are the only options.

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