Question 5
Determine all functions $f:\mathbb R\to\mathbb R$ satisfying
$$f(\sqrt2\,x)+f((4+3\sqrt2)x)=a f((2+\sqrt2)x)\qquad(x\in\mathbb R).$$
The parameter $a$ is retained exactly as printed. The classification below applies to every fixed real $a$; the supplied worked solution instead specializes to coefficient $2$. No continuity, measurability or other regularity is assumed.
$$f(\sqrt2\,x)+f((4+3\sqrt2)x)=a f((2+\sqrt2)x)\qquad(x\in\mathbb R).$$
The parameter $a$ is retained exactly as printed. The classification below applies to every fixed real $a$; the supplied worked solution instead specializes to coefficient $2$. No continuity, measurability or other regularity is assumed.