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Home » But the problem says: "filling into bottles that each hold half of the previous". - Crankk.io

But the problem says: "filling into bottles that each hold half of the previous". - Crankk.io

📅 February 28, 2026 👤 scraface

Feb 28, 2026

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📌 f(ab) = f(a)f(b)

📌 Solution: We are given two functional equations:

📌 $ f(a + b + c) = f(a) + f(b) + f(c) $ for all real $ a, b, c $

📌 $ f(ab) = f(a)f(b) $ for all real $ a, b $

📌 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.

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