Functional Functions that Function 2

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

#Algebra

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`\sum_{i=1}^3`	$\sum_{i=1}^3$
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Comments

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. $\square$

mathh mathh - 6 years, 10 months ago

COOL SOLUTION

A Former Brilliant Member - 6 years, 10 months ago

Math	Appears as
Remember to wrap math in `\(` ... `\)` or `\[` ... `\]` to ensure proper formatting.
`2 \times 3`	$2 \times 3$
`2^{34}`	$2^{34}$
`a_{i-1}`	$a_{i-1}$
`\frac{2}{3}`	$\frac{2}{3}$
`\sqrt{2}`	$\sqrt{2}$
`\sum_{i=1}^3`	$\sum_{i=1}^3$
`\sin \theta`	$\sin \theta$
`\boxed{123}`	$\boxed{123}$