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Home » أولاً، نحول 3 إلى كسر بمقام 4: - 500apps

أولاً، نحول 3 إلى كسر بمقام 4: - 500apps

📅 March 14, 2026 👤 scraface

Mar 14, 2026

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

📌 So $ f $ is both additive and multiplicative. These are very restrictive properties.

📌 Known result: The only functions $ f : \mathbb{R} \to \mathbb{R} $ that are both additive and multiplicative are:

📌 $ f(x) = 0 $ for all $ x $

📌 $ f(x) = x $ for all $ x $

📌 Let’s verify both:

📌 $ 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

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