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# Functional Functions that Function 2

Find all functions $$f : \mathbb{R} \rightarrow \mathbb{R}$$ such that $$f (xy) = x f(x) + y f(y)$$.

Note by Yuxuan Seah
2 years, 2 months ago

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COOL SOLUTION · 2 years, 2 months ago

Let $$P(x,y)$$ be the statement $$f(xy)=xf(x)+yf(y)$$.

$$P(0,0)\implies f(0)=0$$

$$P(x,0)\implies xf(x)=0$$

$$\implies f(x)=\begin{cases}0 && x\neq 0\\ a && a\in\mathbb R, x=0\end{cases}$$

$$P(1,0)\implies a=0$$

Hence $$f(x)=0$$ is the only possible solution and by checking it we see it works. $$\square$$ · 2 years, 2 months ago