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university of pittsburgh--pittsburgh campus - Malaeb

📅 May 4, 2026 👤 scraface

May 04, 2026

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📌 for all real $ x, y $. Find all such functions $ f $.

📌 From additivity, $ f(0) = 0 $, and $ f(-x) = -f(x) $.

📌 If $ f(1) = 1 $, then $ f(n) = n $ for all integers $ n $, and $ f(q) = q $ for all rational $ q $.

📌 Extending to reals under additivity and multiplicativity forces $ f(x) = x $ for all $ x $, by continuity or density of rationals (or using Baire category).

📌 Multiplying both sides by $ n(n+1)(n+2) $:

📌 = A(n+1)(n+2) + B(n)(n+2) + C(n)(n+1).

📌 Subtracting the second equation from the first:

📌 But better: recall earlier expression:

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