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Home » 2025 Kids Movie Games Are ON—Discover the Adventures That Will Keep Kids Talking! - Abbey Badges

2025 Kids Movie Games Are ON—Discover the Adventures That Will Keep Kids Talking! - Abbey Badges

📅 February 27, 2026 👤 scraface

Feb 27, 2026

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📌 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

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

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