2001 Putnam B5

Let $a$ and $b$ be real numbers in the interval $(0,1/2)$, and

let $g$ be a continuous real-valued function such that

$g(g(x))= ag(x)+bx$ for all real $x$. Prove that

$g(x)=cx$ for some constant $c$.

设$a$和$b$为$(0,1/2)$区间内的实数，并且

令 $g$ 为连续实值函数，使得

对于所有真实的 $x$，$g(g(x))= ag(x)+bx$。证明

对于某个常数 $c$，$g(x)=cx$。