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Prove that f(x) lies in [0,1]

Source: All Russian 2014 Grade 10 Day 1 P2

May 3, 2014

functionalgebraRussiaFunctional inequality

Problem Statement

Given a function

f\colon \mathbb{R}\rightarrow \mathbb{R}

with

f(x)^2\le f(y)

for all

x,y\in\mathbb{R}

x>y

, prove that

f(x)\in [0,1]

for all

x\in \mathbb{R}