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📅 May 16, 2026 👤 scraface

May 16, 2026

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📌 $ f(x) = 0 $: $ f(a + b + c) = 0 = 0 + 0 + 0 = f(a) + f(b) + f(c) $, and $ f(ab) = 0 = 0 \cdot 0 = f(a)f(b) $ — OK

📌 $ f(x) = x $: $ f(a + b + c) = a + b + c = f(a) + f(b) + f(c) $, and $ f(ab) = ab = f(a)f(b) $ — OK

📌 Are there any others? Suppose $ f $ is additive and multiplicative. Then $ f(1) = f(1)^2 \Rightarrow f(1) = 0 $ or $ 1 $.

📌 If $ f(1) = 0 $, then $ f(n) = 0 $ for all integers $ n $, and by additivity, $ f(r) = 0 $ for all rational $ r $. If $ f $ is continuous or linear, then $ f(x) = 0 $ or $ f(x) = x $. Without continuity, pathological solutions exist, but the multiplicative condition rules them out unless $ f(x) = 0 $ or $ f(x) = x $.

📌 Hence, only two such functions exist.

📌 Thus, the number of such functions is

📌 Question: Define $ L(u) = u - \frac{u^3}{3} $ for every real number $ u $. If $ n $ is a positive integer, define $ a_n $ by

📌 a_n = \sum_{k=1}^{n} L\left(\frac{k}{n}\right)

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