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!
2026-03-30 16:54:31.1774889671
Find all continuous $f$ that $f(xy) = xf(y) + yf(x)$.
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Hint $\frac{f(xy)}{xy}=\frac{f(y)}{y}+\frac{f(x)}{x}$.
Set $h(y)=\frac{f(y)}{y}$.
Can you finish from here?
here is complete solution.