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L^1 isometry

Source: 2022 Taiwan Mathematics Olympiad

February 19, 2022
functionalgebraTaiwan

Problem Statement

Find all functions f,g:R2→Rf,g:\mathbb{R}^2\to\mathbb{R} satisfying that ∣f(a,b)−f(c,d)∣+∣g(a,b)−g(c,d)∣=∣a−c∣+∣b−d∣|f(a,b)-f(c,d)|+|g(a,b)-g(c,d)|=|a-c|+|b-d| for all real numbers a,b,c,da,b,c,d.
Proposed by usjl
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