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f(x +y) = f(x) +f(y), f(1/x)=f(x)/x^2 (I Soros Olympiad 1994-95 Ukraine R2 11.1

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June 6, 2024
algebrafunctional

Problem Statement

Let the function f:R→Rf:R \to R satisfies the following conditions: 1) for all x,y∈Rx, y\in R, f(x+y)=f(x)+f(y) f(x +y) = f(x) +f(y) 2)f(1)=1 f(1)=1 3) for all x≠0x \ne 0 , f(1/x)=f(x)x2 f(1/x) =\frac{f(x)}{x^2} Prove that for all x∈Rx \in R, f(x)=xf(x) = x.
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