MathDB

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.

A6