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why is my faucet water cloudy - RTA

📅 May 10, 2026 👤 scraface

May 10, 2026

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$ 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 $...
$ f(a + b + c) = c $, and $ f(a) + f(b) + f(c) = c + c + c = 3c $, so...
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 addi...
f(a + b) = f(a) + f(b)
So $ f $ is additive. Also, from the multiplicative condition:

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