2003 IMO Shortlist S04

Find all nondecreasing functions $f: \mathbb{R}\rightarrow\mathbb{R}$ such that
(i) $f(0) = 0, f(1) = 1;$ (ii) $f(a) + f(b) = f(a)f(b) + f(a + b - ab)$ for all real numbers $a, b$ such that $a < 1 < b$ .

*Proposed by A. Di Pisquale & D. Matthews, Australia*

找到所有非减函数 $f: \mathbb{R}\rightarrow\mathbb{R}$ 使得

(i) $f(0) = 0, f(1) = 1;$ (ii) $f(a) + f(b) = f(a)f(b) + f(a + b - ab)$ 对于所有实数 $a, b$ ，使得 $a < 1 < b$ 。

*由澳大利亚 A. Di Pisquale 和 D. Matthews 提出*