MathDB
Continuous from positives to >1

Source: 2024 Israel TST Test 3 P3

January 29, 2024
functional equationalgebracontinuous functionPositive realsfunction

Problem Statement

Find all continuous functions f ⁣:R>0→R≥1f\colon \mathbb{R}_{>0}\to \mathbb{R}_{\geq 1} for which the following equation holds for all positive reals xx, yy: f(f(x)y)−f(f(y)x)=xy(f(x+1)−f(y+1))f\left(\frac{f(x)}{y}\right)-f\left(\frac{f(y)}{x}\right)=xy\left(f(x+1)-f(y+1)\right)
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