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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) $f(f(x)) = a f(x).$
Baltic Way 2016

Functional Equation

Easy
Suggested Book
Functional Equations by BJ Venkatachala

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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