Ngô Quốc Anh

May 1, 2008

Primitives…

Filed under: Giải Tích 2 — Ngô Quốc Anh @ 16:37

Darboux property
RMO 2008, Grade 12, Problem 1


Let and be a continuous function on and having Darboux property on . Prove that if and for all nonnegative we have then admits primitives on .

Proof. The title condition is almost entirely redundant- it simply says that .

Obviously, is an antiderivative away from zero. The only interesting question is the one-sided derivative . Now, \frac{d}{dx}\frac{F(x)}{x}=\frac{xf(x)-F(x)}{x^2}\ge 0 by hypothesis. We also have \liminf{x\to 0}\frac{F(x)}{x}\le \liminf_{x\to 0}f(x)=0 by the same integral inequality. Since is increasing (and bounded below), it has a limit at zero. Since the is zero, this limit is zero, and has the proper one-sided derivative at zero.

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