2008 Putnam B5

Find all continuously differentiable functions $f: \mathbb{R} \to \mathbb{R}$

such that for every rational number $q$, the number $f(q)$ is rational

and has the same denominator as $q$. (The denominator of a rational number

$q$ is the unique positive integer $b$ such that $q = a/b$

for some integer $a$ with $\mathrm{gcd}(a,b) = 1$.)

(Note: gcd means greatest common

divisor.)

查找所有连续可微函数 $f: \mathbb{R} \to \mathbb{R}$

这样对于每个有理数 $q$，数字 $f(q)$ 都是有理数

并且与 $q$ 具有相同的分母。 (有理数的分母

$q$ 是唯一正整数 $b$，使得 $q = a/b$

对于某个整数 $a$ 且 $\mathrm{gcd}(a,b) = 1$。)

(注：gcd 表示最大常见

除数。)