2019 Putnam A6

Let $g$ be a real-valued function that is continuous on the closed interval $[0,1]$ and twice differentiable on

the open interval $(0,1)$. Suppose that for some real number $r>1$,

$$

\lim_{x \to 0^+} \frac{g(x)}{x^r} = 0.

$$

Prove that either

$$

\lim_{x \to 0^+} g'(x) = 0 \qquad \text{or} \qquad \limsup_{x \to 0^+} x^r |g''(x)| = \infty.

$$

设 $g$ 为实值函数，在闭区间 $[0,1]$ 上连续，并且在

开区间 $(0,1)$。假设对于某个实数 $r>1$，

$$

\lim_{x \to 0^+} \frac{g(x)}{x^r} = 0。

$$

证明无论是

$$

\lim_{x \to 0^+} g'(x) = 0 \qquad \text{或} \qquad \limsup_{x \to 0^+} x^r |g''(x)| = \infty。

$$