📝 题目
例 27 设 $f\left( x\right)$ 在 $\left( {0, + \infty }\right)$ 上连续,且满足条件
$$ f\left( {x}^{2}\right) = f\left( x\right) \;\left( {\forall x > 0}\right) . $$
求证: $f\left( x\right)$ 为一常数.
💡 答案与解析
证 由条件得
$$ f\left( x\right) = f\left( {x}^{1/2}\right) = f\left( {x}^{1/{2}^{2}}\right) = \cdots = f\left( {x}^{1/{2}^{n}}\right) = \cdots $$
$$ \Rightarrow f\left( x\right) = \mathop{\lim }\limits_{{n \rightarrow \infty }}f\left( {x}^{\frac{1}{{2}^{n}}}\right) = f\left( {\mathop{\lim }\limits_{{n \rightarrow \infty }}{x}^{\frac{1}{{2}^{n}}}}\right) = f\left( 1\right) , $$
即
$$ f\left( x\right) \equiv f\left( 1\right) \text{ . } $$