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B) $ 3\sqrt3s - 12\sqrt3 $ - Portal da AcÃºstica

📅 March 1, 2026 👤 scraface

Mar 01, 2026

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📌 for all $ a, b, c \in \mathbb{R} $.

📌 f(a + b + c) = f(a) + f(b) + f(c) \quad \forall a, b, c \in \mathbb{R}

📌 Let’s set $ a = b = c = 0 $:

📌 f(0) = 3f(0) \Rightarrow 2f(0) = 0 \Rightarrow f(0) = 0

📌 Now, set $ c = 0 $:

📌 This is the **Cauchy Functional Equation**, whose solutions over $ \mathbb{R} $ are linear functions $ f(x) = kx $, assuming some regularity condition (like continuity, monotonicity, or boundedness). Since the problem does not restrict $ f $, but asks for the **number** of such functions, we must consider all additive functions from $ \mathbb{R} \to \mathbb{R} $.

📌 Under the Axiom of Choice, there are **infinitely many** additive functions (including pathological ones), but if we restrict to **real-valued additive functions** (without further constraints), the number of such functions is uncountably infinite.

📌 However, in the context of a math olympiad, and given the phrasing find the number of functions, we assume the standard assumption of **linear solutions**, i.e., $ f(x) = kx $.

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