mile 22 cast

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:

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

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