# Find all continuous $f$ that $f(xy) = xf(y) + yf(x)$.

Determine all continuous $$f : \mathbb R \rightarrow \mathbb R$$ that satisfies $$f(xy) = xf(y) + yf(x)$$

I tried rewrite the equation as $$f(xy) + f(x)f(y) + xy = (f(x) + x)(f(y) + y)$$ and I know that $$f(0) = f(1) = 0$$. Thanks in advance!

• The equation looks a lot like the product rule.......$f(x)$ could be something related to the derivative operator.......Maybe : $f(x)=\frac{dx}{dt}$ ($t$ be some dummy variable) – Saket Gurjar May 30 '20 at 10:25
• See artofproblemsolving.com/community/c6h1667141p10595936. The idea is $g(x)=f(x)/x$ for $x\neq 0$ gives $g(xy)=g(x)+g(y)$. Then $h(x)=g(e^x)$ gives $h(x+y)=h(x)+h(y)$ (Cauchy's functional equation). Also $f(x) \equiv 0$ is an obvious solution. – Sil May 30 '20 at 10:30

Hint $$\frac{f(xy)}{xy}=\frac{f(y)}{y}+\frac{f(x)}{x}$$.
Set $$h(y)=\frac{f(y)}{y}$$.