Ngô Quốc Anh

May 1, 2008

Nondecreasing function

Filed under: Các Bài Tập Nhỏ, Giải Tích 1 — Ngô Quốc Anh @ 16:39

Nice nondecreasing function
RMO 2008, 11th Grade, Problem 1

Let be a continous function such that the sequences are nondecreasing for any real number . Prove that is nondecreasing.

Proof. It’s trivial too see that if , then f\left( \frac mn x \right) \geq f(x), for all , so in particular for all rational numbers . Now let , and let , . If , let . Because is continuous, there exists a rational number , such that f(x_0)= f(y) + \frac {\delta}{2}. But , so contradiction. This means that if , and , such that then .

Now let be any two real numbers. Then there exists such that , and using the above claim it follows that which proves that is nondecreasing.

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