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📅 March 10, 2026 👤 scraface

Mar 10, 2026

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📌 We aim to find all such functions $ f : \mathbb{R} \to \mathbb{R} $.

📌 Let’s try constant functions first. Suppose $ f(x) = c $ for all $ x $. Then:

📌 $ f(a + b + c) = c $, and $ f(a) + f(b) + f(c) = c + c + c = 3c $, so $ c = 3c \Rightarrow c = 0 $

📌 Also, $ f(ab) = 0 = 0 \cdot 0 = f(a)f(b) $, so the zero function works.

📌 Now suppose $ f $ is not constant.

📌 Let’s try additive functions satisfying both properties. From the additive condition, set $ c = 0 $:

📌 f(a + b) = f(a) + f(b)

📌 So $ f $ is additive. Also, from the multiplicative condition:

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