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} $.

Sun, 03 May 2026 18:59:33 +0000

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