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,
x→0+limxrg(x)=0.
Prove that either
x→0+limg′(x)=0orx→0+limsupxr∣g′′(x)∣=∞.