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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